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5c60c468a1d7266eeaebea1ff88d161a4dead750..8e5b414df9239d9397a4308f84d2e183de360406 100644 GIT binary patch delta 160 zcmbQTlx50NmJOOh3Kk}*iRR`8hN*^z$tI~OMwY3`#);+z7O94oiAl+cMw2~-77PgX35c1u?-5}+e?-B+B-z5qBrzFipMi;iu~A~Oky%Qzxq-1| zqJe>dMbh-uH(90=RJA?n7KKR#Kvbw*7@**pk^!IUg! 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Source code for geomstats.algebra_utils

 """Utility module of reusable algebra routines."""
+
 import math
 
 import geomstats.backend as gs
diff --git a/_modules/geomstats/datasets/prepare_emg_data.html b/_modules/geomstats/datasets/prepare_emg_data.html
index b8e5d9686..efab86071 100644
--- a/_modules/geomstats/datasets/prepare_emg_data.html
+++ b/_modules/geomstats/datasets/prepare_emg_data.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/datasets/prepare_graph_data.html b/_modules/geomstats/datasets/prepare_graph_data.html
index b3917c92e..143c3ddce 100644
--- a/_modules/geomstats/datasets/prepare_graph_data.html
+++ b/_modules/geomstats/datasets/prepare_graph_data.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/datasets/utils.html b/_modules/geomstats/datasets/utils.html
index e8335edff..76e827f87 100644
--- a/_modules/geomstats/datasets/utils.html
+++ b/_modules/geomstats/datasets/utils.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/distributions/lognormal.html b/_modules/geomstats/distributions/lognormal.html
index fb0949022..ebb7a3c9a 100644
--- a/_modules/geomstats/distributions/lognormal.html
+++ b/_modules/geomstats/distributions/lognormal.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/errors.html b/_modules/geomstats/errors.html
index 75df30049..0dab45355 100644
--- a/_modules/geomstats/errors.html
+++ b/_modules/geomstats/errors.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/exceptions.html b/_modules/geomstats/exceptions.html
index db1a71436..df9566a8d 100644
--- a/_modules/geomstats/exceptions.html
+++ b/_modules/geomstats/exceptions.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/base.html b/_modules/geomstats/geometry/base.html
index 7d004f1c3..36633e6b7 100644
--- a/_modules/geomstats/geometry/base.html
+++ b/_modules/geomstats/geometry/base.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/complex_manifold.html b/_modules/geomstats/geometry/complex_manifold.html
index e09854382..614f66def 100644
--- a/_modules/geomstats/geometry/complex_manifold.html
+++ b/_modules/geomstats/geometry/complex_manifold.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/complex_matrices.html b/_modules/geomstats/geometry/complex_matrices.html
index d27e863c0..a0daf94af 100644
--- a/_modules/geomstats/geometry/complex_matrices.html
+++ b/_modules/geomstats/geometry/complex_matrices.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/complex_poincare_disk.html b/_modules/geomstats/geometry/complex_poincare_disk.html
index 69d52f033..d26c0b93e 100644
--- a/_modules/geomstats/geometry/complex_poincare_disk.html
+++ b/_modules/geomstats/geometry/complex_poincare_disk.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/complex_riemannian_metric.html b/_modules/geomstats/geometry/complex_riemannian_metric.html
index e54be57bb..13bd1c8ae 100644
--- a/_modules/geomstats/geometry/complex_riemannian_metric.html
+++ b/_modules/geomstats/geometry/complex_riemannian_metric.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/connection.html b/_modules/geomstats/geometry/connection.html
index f6001c76f..686d11787 100644
--- a/_modules/geomstats/geometry/connection.html
+++ b/_modules/geomstats/geometry/connection.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/diffeo.html b/_modules/geomstats/geometry/diffeo.html
index b930923ed..3cf9a9dec 100644
--- a/_modules/geomstats/geometry/diffeo.html
+++ b/_modules/geomstats/geometry/diffeo.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/discrete_curves.html b/_modules/geomstats/geometry/discrete_curves.html
index 93bcdac7d..4991b39f8 100644
--- a/_modules/geomstats/geometry/discrete_curves.html
+++ b/_modules/geomstats/geometry/discrete_curves.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/discrete_surfaces.html b/_modules/geomstats/geometry/discrete_surfaces.html
index 47fc02e4d..edde50016 100644
--- a/_modules/geomstats/geometry/discrete_surfaces.html
+++ b/_modules/geomstats/geometry/discrete_surfaces.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -432,6 +432,7 @@ 
Source code for geomstats.geometry.discrete_surfaces
Lead authors: Emmanuel Hartman, Adele Myers.
 """
+
 import math
 
 import geomstats.backend as gs
diff --git a/_modules/geomstats/geometry/euclidean.html b/_modules/geomstats/geometry/euclidean.html
index 58aac119d..d4d250e82 100644
--- a/_modules/geomstats/geometry/euclidean.html
+++ b/_modules/geomstats/geometry/euclidean.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/fiber_bundle.html b/_modules/geomstats/geometry/fiber_bundle.html
index 11bbe50b5..85dfdb6e7 100644
--- a/_modules/geomstats/geometry/fiber_bundle.html
+++ b/_modules/geomstats/geometry/fiber_bundle.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/full_rank_correlation_matrices.html b/_modules/geomstats/geometry/full_rank_correlation_matrices.html
index 9bba6bbd6..86764560c 100644
--- a/_modules/geomstats/geometry/full_rank_correlation_matrices.html
+++ b/_modules/geomstats/geometry/full_rank_correlation_matrices.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/full_rank_matrices.html b/_modules/geomstats/geometry/full_rank_matrices.html
index 2bbdc8045..4ca880cba 100644
--- a/_modules/geomstats/geometry/full_rank_matrices.html
+++ b/_modules/geomstats/geometry/full_rank_matrices.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/functions.html b/_modules/geomstats/geometry/functions.html
index fd9ad0d23..711e86cb0 100644
--- a/_modules/geomstats/geometry/functions.html
+++ b/_modules/geomstats/geometry/functions.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/general_linear.html b/_modules/geomstats/geometry/general_linear.html
index 04f9a5989..7b91c2f6a 100644
--- a/_modules/geomstats/geometry/general_linear.html
+++ b/_modules/geomstats/geometry/general_linear.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/grassmannian.html b/_modules/geomstats/geometry/grassmannian.html
index 9e38534af..9d64f4135 100644
--- a/_modules/geomstats/geometry/grassmannian.html
+++ b/_modules/geomstats/geometry/grassmannian.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/heisenberg.html b/_modules/geomstats/geometry/heisenberg.html
index 0624ec94c..899a836ae 100644
--- a/_modules/geomstats/geometry/heisenberg.html
+++ b/_modules/geomstats/geometry/heisenberg.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/hermitian.html b/_modules/geomstats/geometry/hermitian.html
index 40735858e..f886d8077 100644
--- a/_modules/geomstats/geometry/hermitian.html
+++ b/_modules/geomstats/geometry/hermitian.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/hermitian_matrices.html b/_modules/geomstats/geometry/hermitian_matrices.html
index 80bc238ff..6e0d195f8 100644
--- a/_modules/geomstats/geometry/hermitian_matrices.html
+++ b/_modules/geomstats/geometry/hermitian_matrices.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -432,6 +432,7 @@ 
Source code for geomstats.geometry.hermitian_matrices
Lead author: Yann Cabanes.
 """
+
 import logging
 
 import geomstats.backend as gs
diff --git a/_modules/geomstats/geometry/hpd_matrices.html b/_modules/geomstats/geometry/hpd_matrices.html
index 28fe9d80b..d847cff7e 100644
--- a/_modules/geomstats/geometry/hpd_matrices.html
+++ b/_modules/geomstats/geometry/hpd_matrices.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/hyperbolic.html b/_modules/geomstats/geometry/hyperbolic.html
index 37f1f5b7e..b7b33b4bf 100644
--- a/_modules/geomstats/geometry/hyperbolic.html
+++ b/_modules/geomstats/geometry/hyperbolic.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/hyperboloid.html b/_modules/geomstats/geometry/hyperboloid.html
index c25a35f0b..50de986c3 100644
--- a/_modules/geomstats/geometry/hyperboloid.html
+++ b/_modules/geomstats/geometry/hyperboloid.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -763,12 +763,8 @@ 
Source code for geomstats.geometry.hyperboloid
        """
         angle = self.dist(base_point, point)
 
-        coef_1_ = utils.taylor_exp_even_func(
-            angle**2, utils.inv_sinch_close_0, order=4
-        )
-        coef_2_ = utils.taylor_exp_even_func(
-            angle**2, utils.inv_tanh_close_0, order=4
-        )
+        coef_1_ = utils.taylor_exp_even_func(angle**2, utils.inv_sinch_close_0, order=4)
+        coef_2_ = utils.taylor_exp_even_func(angle**2, utils.inv_tanh_close_0, order=4)
 
         log_term_1 = gs.einsum("...,...j->...j", coef_1_, point)
         log_term_2 = -gs.einsum("...,...j->...j", coef_2_, base_point)
diff --git a/_modules/geomstats/geometry/hypersphere.html b/_modules/geomstats/geometry/hypersphere.html
index 9a9efdd60..09e1e6ac4 100644
--- a/_modules/geomstats/geometry/hypersphere.html
+++ b/_modules/geomstats/geometry/hypersphere.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/invariant_metric.html b/_modules/geomstats/geometry/invariant_metric.html
index a5ab859d5..709c368c3 100644
--- a/_modules/geomstats/geometry/invariant_metric.html
+++ b/_modules/geomstats/geometry/invariant_metric.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/landmarks.html b/_modules/geomstats/geometry/landmarks.html
index 2e553e3e4..a96efcb7a 100644
--- a/_modules/geomstats/geometry/landmarks.html
+++ b/_modules/geomstats/geometry/landmarks.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/lie_algebra.html b/_modules/geomstats/geometry/lie_algebra.html
index 386816379..0027e45b4 100644
--- a/_modules/geomstats/geometry/lie_algebra.html
+++ b/_modules/geomstats/geometry/lie_algebra.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/lie_group.html b/_modules/geomstats/geometry/lie_group.html
index 19932c35f..a10f62a41 100644
--- a/_modules/geomstats/geometry/lie_group.html
+++ b/_modules/geomstats/geometry/lie_group.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/lower_triangular_matrices.html b/_modules/geomstats/geometry/lower_triangular_matrices.html
index 1b5d53fcf..0e1c3130b 100644
--- a/_modules/geomstats/geometry/lower_triangular_matrices.html
+++ b/_modules/geomstats/geometry/lower_triangular_matrices.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/manifold.html b/_modules/geomstats/geometry/manifold.html
index c6b52009a..f1de68ff1 100644
--- a/_modules/geomstats/geometry/manifold.html
+++ b/_modules/geomstats/geometry/manifold.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/matrices.html b/_modules/geomstats/geometry/matrices.html
index b9f625ad1..270033d6d 100644
--- a/_modules/geomstats/geometry/matrices.html
+++ b/_modules/geomstats/geometry/matrices.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -429,6 +429,7 @@
                   
   
Source code for geomstats.geometry.matrices
 """Module exposing the `Matrices` and `MatricesMetric` class."""
+
 import logging
 import math
 from functools import reduce
diff --git a/_modules/geomstats/geometry/minkowski.html b/_modules/geomstats/geometry/minkowski.html
index 1a2d2cb0f..17b68cfe6 100644
--- a/_modules/geomstats/geometry/minkowski.html
+++ b/_modules/geomstats/geometry/minkowski.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/nfold_manifold.html b/_modules/geomstats/geometry/nfold_manifold.html
index 1e2becc41..b6558e9fd 100644
--- a/_modules/geomstats/geometry/nfold_manifold.html
+++ b/_modules/geomstats/geometry/nfold_manifold.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -433,7 +433,6 @@ 
Source code for geomstats.geometry.nfold_manifold
Lead author: Nicolas Guigui, John Harvey.
 """
 
-
 import geomstats.backend as gs
 import geomstats.errors
 from geomstats.geometry.manifold import Manifold
diff --git a/_modules/geomstats/geometry/open_hemisphere.html b/_modules/geomstats/geometry/open_hemisphere.html
index 79f8d50bb..609ed00a4 100644
--- a/_modules/geomstats/geometry/open_hemisphere.html
+++ b/_modules/geomstats/geometry/open_hemisphere.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/poincare_ball.html b/_modules/geomstats/geometry/poincare_ball.html
index d3337d673..dc4769fa0 100644
--- a/_modules/geomstats/geometry/poincare_ball.html
+++ b/_modules/geomstats/geometry/poincare_ball.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/poincare_half_space.html b/_modules/geomstats/geometry/poincare_half_space.html
index a05459668..60f2a75aa 100644
--- a/_modules/geomstats/geometry/poincare_half_space.html
+++ b/_modules/geomstats/geometry/poincare_half_space.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/poincare_polydisk.html b/_modules/geomstats/geometry/poincare_polydisk.html
index aedd69c58..4ad4390a9 100644
--- a/_modules/geomstats/geometry/poincare_polydisk.html
+++ b/_modules/geomstats/geometry/poincare_polydisk.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/positive_lower_triangular_matrices.html b/_modules/geomstats/geometry/positive_lower_triangular_matrices.html
index a6c5bf119..f1ca85baf 100644
--- a/_modules/geomstats/geometry/positive_lower_triangular_matrices.html
+++ b/_modules/geomstats/geometry/positive_lower_triangular_matrices.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/positive_reals.html b/_modules/geomstats/geometry/positive_reals.html
index bbef67bc9..ffc6221da 100644
--- a/_modules/geomstats/geometry/positive_reals.html
+++ b/_modules/geomstats/geometry/positive_reals.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/pre_shape.html b/_modules/geomstats/geometry/pre_shape.html
index 2c12e43a1..bacd8b82c 100644
--- a/_modules/geomstats/geometry/pre_shape.html
+++ b/_modules/geomstats/geometry/pre_shape.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/product_hpd_and_siegel_disks.html b/_modules/geomstats/geometry/product_hpd_and_siegel_disks.html
index 20fe0e0d7..e37cefcad 100644
--- a/_modules/geomstats/geometry/product_hpd_and_siegel_disks.html
+++ b/_modules/geomstats/geometry/product_hpd_and_siegel_disks.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/product_manifold.html b/_modules/geomstats/geometry/product_manifold.html
index dcfffdd51..d8070aa82 100644
--- a/_modules/geomstats/geometry/product_manifold.html
+++ b/_modules/geomstats/geometry/product_manifold.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -432,6 +432,7 @@ 
Source code for geomstats.geometry.product_manifold
Lead author: Nicolas Guigui, John Harvey.
 """
+
 import math
 
 import geomstats.backend as gs
@@ -731,9 +732,11 @@ 
Source code for geomstats.geometry.product_manifold
if default_coords_type == "extrinsic":
             factor_embedding_spaces = [
-                manifold.embedding_space
-                if hasattr(manifold, "embedding_space")
-                else manifold
+                (
+                    manifold.embedding_space
+                    if hasattr(manifold, "embedding_space")
+                    else manifold
+                )
                 for manifold in factors
             ]
             # TODO: need to revisit due to removal of scales
diff --git a/_modules/geomstats/geometry/product_positive_reals_and_poincare_disks.html b/_modules/geomstats/geometry/product_positive_reals_and_poincare_disks.html
index df26c9064..2155714c2 100644
--- a/_modules/geomstats/geometry/product_positive_reals_and_poincare_disks.html
+++ b/_modules/geomstats/geometry/product_positive_reals_and_poincare_disks.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -475,6 +475,7 @@ 
Source code for geomstats.geometry.product_positive_reals_and_poincare_disks
 .. [Yang_2013] Marc Arnaudon, Frédéric Barbaresco and Le Yang. Riemannian Medians
     and Means With Applications to Radar Signal Processing, IEEE, 2013.
 """
+
 from geomstats.geometry.complex_poincare_disk import (
     ComplexPoincareDisk,
     ComplexPoincareDiskMetric,
diff --git a/_modules/geomstats/geometry/pullback_metric.html b/_modules/geomstats/geometry/pullback_metric.html
index 276ae7add..b3bfb7032 100644
--- a/_modules/geomstats/geometry/pullback_metric.html
+++ b/_modules/geomstats/geometry/pullback_metric.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -432,6 +432,7 @@ 
Source code for geomstats.geometry.pullback_metric
Lead author: Nina Miolane.
 """
+
 import geomstats.backend as gs
 from geomstats.geometry.riemannian_metric import RiemannianMetric
 from geomstats.numerics.geodesic import ExpODESolver, LogShootingSolver
diff --git a/_modules/geomstats/geometry/quotient_metric.html b/_modules/geomstats/geometry/quotient_metric.html
index 2f7280bad..7c1d94912 100644
--- a/_modules/geomstats/geometry/quotient_metric.html
+++ b/_modules/geomstats/geometry/quotient_metric.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/rank_k_psd_matrices.html b/_modules/geomstats/geometry/rank_k_psd_matrices.html
index f8cd144c8..162c299da 100644
--- a/_modules/geomstats/geometry/rank_k_psd_matrices.html
+++ b/_modules/geomstats/geometry/rank_k_psd_matrices.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/riemannian_metric.html b/_modules/geomstats/geometry/riemannian_metric.html
index b0a2361f2..8f73c5054 100644
--- a/_modules/geomstats/geometry/riemannian_metric.html
+++ b/_modules/geomstats/geometry/riemannian_metric.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -432,6 +432,7 @@ 
Source code for geomstats.geometry.riemannian_metric
Lead author: Nina Miolane.
 """
+
 from abc import ABC
 
 import joblib
diff --git a/_modules/geomstats/geometry/sasaki_metric.html b/_modules/geomstats/geometry/sasaki_metric.html
index 03a7ef19b..241321fc9 100644
--- a/_modules/geomstats/geometry/sasaki_metric.html
+++ b/_modules/geomstats/geometry/sasaki_metric.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -435,6 +435,7 @@ 
Source code for geomstats.geometry.sasaki_metric

 
 Lead authors: E. Nava-Yazdani, F. Ambellan, M. Hanik and C. von Tycowicz.
 """
+
 from joblib import Parallel, delayed
 
 import geomstats.backend as gs
diff --git a/_modules/geomstats/geometry/scalar_product_metric.html b/_modules/geomstats/geometry/scalar_product_metric.html
index e4444855b..e561c2335 100644
--- a/_modules/geomstats/geometry/scalar_product_metric.html
+++ b/_modules/geomstats/geometry/scalar_product_metric.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -434,6 +434,7 @@ 
Source code for geomstats.geometry.scalar_product_metric
Lead author: John Harvey.
 """
+
 from functools import wraps
 
 import geomstats.backend as gs
diff --git a/_modules/geomstats/geometry/siegel.html b/_modules/geomstats/geometry/siegel.html
index 7c7b63880..bb18a6cfe 100644
--- a/_modules/geomstats/geometry/siegel.html
+++ b/_modules/geomstats/geometry/siegel.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/skew_symmetric_matrices.html b/_modules/geomstats/geometry/skew_symmetric_matrices.html
index d6b2d62e5..1a404d158 100644
--- a/_modules/geomstats/geometry/skew_symmetric_matrices.html
+++ b/_modules/geomstats/geometry/skew_symmetric_matrices.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/spd_matrices.html b/_modules/geomstats/geometry/spd_matrices.html
index 5b34b299b..f8db439b7 100644
--- a/_modules/geomstats/geometry/spd_matrices.html
+++ b/_modules/geomstats/geometry/spd_matrices.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/special_euclidean.html b/_modules/geomstats/geometry/special_euclidean.html
index b4e233df7..2b821a52f 100644
--- a/_modules/geomstats/geometry/special_euclidean.html
+++ b/_modules/geomstats/geometry/special_euclidean.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -434,6 +434,7 @@ 
Source code for geomstats.geometry.special_euclidean
Lead authors: Nicolas Guigui and Nina Miolane.
 """
+
 import math
 
 import geomstats.algebra_utils as utils
@@ -1315,10 +1316,7 @@ 
Source code for geomstats.geometry.special_euclidean
coef_2 = gs.zeros_like(angle)
 
         coef_2 += mask_close_0_float * (
-            1.0 / 12.0
-            + angle**2 / 720.0
-            + angle**4 / 30240.0
-            + angle**6 / 1209600.0
+            1.0 / 12.0 + angle**2 / 720.0 + angle**4 / 30240.0 + angle**6 / 1209600.0
         )
 
         delta_angle = angle - gs.pi
diff --git a/_modules/geomstats/geometry/special_orthogonal.html b/_modules/geomstats/geometry/special_orthogonal.html
index 666ae3836..c652f5d34 100644
--- a/_modules/geomstats/geometry/special_orthogonal.html
+++ b/_modules/geomstats/geometry/special_orthogonal.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/stiefel.html b/_modules/geomstats/geometry/stiefel.html
index d040cb6c6..5408c78a3 100644
--- a/_modules/geomstats/geometry/stiefel.html
+++ b/_modules/geomstats/geometry/stiefel.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/stratified/graph_space.html b/_modules/geomstats/geometry/stratified/graph_space.html
index d14c95ba6..a73d456eb 100644
--- a/_modules/geomstats/geometry/stratified/graph_space.html
+++ b/_modules/geomstats/geometry/stratified/graph_space.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/stratified/point_set.html b/_modules/geomstats/geometry/stratified/point_set.html
index b696a92d9..f22cbb775 100644
--- a/_modules/geomstats/geometry/stratified/point_set.html
+++ b/_modules/geomstats/geometry/stratified/point_set.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/stratified/spider.html b/_modules/geomstats/geometry/stratified/spider.html
index 59110fce5..b78ab3e71 100644
--- a/_modules/geomstats/geometry/stratified/spider.html
+++ b/_modules/geomstats/geometry/stratified/spider.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -769,9 +769,11 @@ 
Source code for geomstats.geometry.stratified.spider
)
             x = g(t)
             return [
-                SpiderPoint(stratum=initial_point.stratum, stratum_coord=-xx[0])
-                if xx < 0.0
-                else SpiderPoint(stratum=end_point.stratum, stratum_coord=xx[0])
+                (
+                    SpiderPoint(stratum=initial_point.stratum, stratum_coord=-xx[0])
+                    if xx < 0.0
+                    else SpiderPoint(stratum=end_point.stratum, stratum_coord=xx[0])
+                )
                 for xx in x
             ]
 
diff --git a/_modules/geomstats/geometry/stratified/wald_space.html b/_modules/geomstats/geometry/stratified/wald_space.html
index 0b8f4d2ed..716c45b37 100644
--- a/_modules/geomstats/geometry/stratified/wald_space.html
+++ b/_modules/geomstats/geometry/stratified/wald_space.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/sub_riemannian_metric.html b/_modules/geomstats/geometry/sub_riemannian_metric.html
index cbc38e565..0b5e2b1d0 100644
--- a/_modules/geomstats/geometry/sub_riemannian_metric.html
+++ b/_modules/geomstats/geometry/sub_riemannian_metric.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/geometry/symmetric_matrices.html b/_modules/geomstats/geometry/symmetric_matrices.html
index e83fef419..a68b4d929 100644
--- a/_modules/geomstats/geometry/symmetric_matrices.html
+++ b/_modules/geomstats/geometry/symmetric_matrices.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/information_geometry/base.html b/_modules/geomstats/information_geometry/base.html
index 903ead7aa..1a27d4ab1 100644
--- a/_modules/geomstats/information_geometry/base.html
+++ b/_modules/geomstats/information_geometry/base.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/information_geometry/beta.html b/_modules/geomstats/information_geometry/beta.html
index 1fea53e00..cc57d870e 100644
--- a/_modules/geomstats/information_geometry/beta.html
+++ b/_modules/geomstats/information_geometry/beta.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/information_geometry/binomial.html b/_modules/geomstats/information_geometry/binomial.html
index 59625e7e4..fb1a9e71d 100644
--- a/_modules/geomstats/information_geometry/binomial.html
+++ b/_modules/geomstats/information_geometry/binomial.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -432,6 +432,7 @@ 
Source code for geomstats.information_geometry.binomial
Lead authors: Jules Deschamps, Tra My Nguyen.
 """
+
 from scipy.special import factorial
 from scipy.stats import binom
 
@@ -611,9 +612,7 @@ 
Source code for geomstats.information_geometry.binomial
factorial(k) * factorial(self.n_draws - k)
             )
             return (
-                gs.from_numpy(const)
-                * (point**k)
-                * ((1 - point) ** (self.n_draws - k))
+                gs.from_numpy(const) * (point**k) * ((1 - point) ** (self.n_draws - k))
             )
 
         return pmf

diff --git a/_modules/geomstats/information_geometry/categorical.html b/_modules/geomstats/information_geometry/categorical.html
index 5da78972c..227f01e30 100644
--- a/_modules/geomstats/information_geometry/categorical.html
+++ b/_modules/geomstats/information_geometry/categorical.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/information_geometry/dirichlet.html b/_modules/geomstats/information_geometry/dirichlet.html
index 396f0c27e..327ab1932 100644
--- a/_modules/geomstats/information_geometry/dirichlet.html
+++ b/_modules/geomstats/information_geometry/dirichlet.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -432,6 +432,7 @@ 
Source code for geomstats.information_geometry.dirichlet
Lead author: Alice Le Brigant.
 """
+
 import math
 
 import numpy as np
@@ -764,9 +765,7 @@ 
Source code for geomstats.information_geometry.dirichlet
jac_1 = term_1 * term_8 / term_9
         jac_1_mat = gs.squeeze(gs.tile(jac_1, (dim, dim, dim, 1, 1)))
         jac_2 = (
-            -term_6
-            / term_9**2
-            * gs.einsum("j...,i...->ji...", term_4 - term_3, term_1)
+            -term_6 / term_9**2 * gs.einsum("j...,i...->ji...", term_4 - term_3, term_1)
         )
         jac_2_mat = gs.squeeze(gs.tile(jac_2, (dim, dim, 1, 1, 1)))
         jac_3 = term_3 * term_6 / term_9
diff --git a/_modules/geomstats/information_geometry/exponential.html b/_modules/geomstats/information_geometry/exponential.html
index 6805460bd..71ebf7243 100644
--- a/_modules/geomstats/information_geometry/exponential.html
+++ b/_modules/geomstats/information_geometry/exponential.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/information_geometry/fisher_rao_metric.html b/_modules/geomstats/information_geometry/fisher_rao_metric.html
index e8a641dc6..1402ded7a 100644
--- a/_modules/geomstats/information_geometry/fisher_rao_metric.html
+++ b/_modules/geomstats/information_geometry/fisher_rao_metric.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/information_geometry/gamma.html b/_modules/geomstats/information_geometry/gamma.html
index 7999e01a3..1e40d5b2a 100644
--- a/_modules/geomstats/information_geometry/gamma.html
+++ b/_modules/geomstats/information_geometry/gamma.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -445,6 +445,7 @@ 
Source code for geomstats.information_geometry.gamma
Lead author: Jules Deschamps.
 """
+
 from scipy.stats import gamma
 
 import geomstats.backend as gs
diff --git a/_modules/geomstats/information_geometry/geometric.html b/_modules/geomstats/information_geometry/geometric.html
index b480b5b6b..a005eaaee 100644
--- a/_modules/geomstats/information_geometry/geometric.html
+++ b/_modules/geomstats/information_geometry/geometric.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/information_geometry/multinomial.html b/_modules/geomstats/information_geometry/multinomial.html
index 6da952d48..25ee0de62 100644
--- a/_modules/geomstats/information_geometry/multinomial.html
+++ b/_modules/geomstats/information_geometry/multinomial.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/information_geometry/normal.html b/_modules/geomstats/information_geometry/normal.html
index a31074dd1..53f5d918d 100644
--- a/_modules/geomstats/information_geometry/normal.html
+++ b/_modules/geomstats/information_geometry/normal.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/information_geometry/poisson.html b/_modules/geomstats/information_geometry/poisson.html
index 87e8d6c4b..52e106acb 100644
--- a/_modules/geomstats/information_geometry/poisson.html
+++ b/_modules/geomstats/information_geometry/poisson.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/integrator.html b/_modules/geomstats/integrator.html
index 452ccc37b..18d77ba99 100644
--- a/_modules/geomstats/integrator.html
+++ b/_modules/geomstats/integrator.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/_template.html b/_modules/geomstats/learning/_template.html
index 7a01719ed..2bb1b6b87 100644
--- a/_modules/geomstats/learning/_template.html
+++ b/_modules/geomstats/learning/_template.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/aac.html b/_modules/geomstats/learning/aac.html
index ed4dd25f9..c05afad1d 100644
--- a/_modules/geomstats/learning/aac.html
+++ b/_modules/geomstats/learning/aac.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/agglomerative_hierarchical_clustering.html b/_modules/geomstats/learning/agglomerative_hierarchical_clustering.html
index 195e41ac9..724d92e9e 100644
--- a/_modules/geomstats/learning/agglomerative_hierarchical_clustering.html
+++ b/_modules/geomstats/learning/agglomerative_hierarchical_clustering.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/expectation_maximization.html b/_modules/geomstats/learning/expectation_maximization.html
index ff78f915a..0f607e187 100644
--- a/_modules/geomstats/learning/expectation_maximization.html
+++ b/_modules/geomstats/learning/expectation_maximization.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -671,9 +671,7 @@ 
Source code for geomstats.learning.expectation_maximization
variances_units = gs.expand_dims(self.variances, 0)
         variances_units = gs.repeat(variances_units, distance_to_mean.shape[0], axis=0)
 
-        distribution_normal = gs.exp(
-            -(distance_to_mean**2) / (2 * variances_units**2)
-        )
+        distribution_normal = gs.exp(-(distance_to_mean**2) / (2 * variances_units**2))
 
         zeta_sigma = (2 * gs.pi) ** (2 / 3) * self.variances
         zeta_sigma = zeta_sigma * gs.exp(
diff --git a/_modules/geomstats/learning/exponential_barycenter.html b/_modules/geomstats/learning/exponential_barycenter.html
index 0f7c0589b..f1b3f29b6 100644
--- a/_modules/geomstats/learning/exponential_barycenter.html
+++ b/_modules/geomstats/learning/exponential_barycenter.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/frechet_mean.html b/_modules/geomstats/learning/frechet_mean.html
index 2b7506307..4f64a9b10 100644
--- a/_modules/geomstats/learning/frechet_mean.html
+++ b/_modules/geomstats/learning/frechet_mean.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -795,8 +795,7 @@ 
Source code for geomstats.learning.frechet_mean
<
         )
 
         while (
-            sq_norm_current_tangent_mean > self.epsilon**2
-            and iteration < self.max_iter
+            sq_norm_current_tangent_mean > self.epsilon**2 and iteration < self.max_iter
         ):
             iteration += 1
 
diff --git a/_modules/geomstats/learning/geodesic_regression.html b/_modules/geomstats/learning/geodesic_regression.html
index 568f9de08..cb79f376c 100644
--- a/_modules/geomstats/learning/geodesic_regression.html
+++ b/_modules/geomstats/learning/geodesic_regression.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/geometric_median.html b/_modules/geomstats/learning/geometric_median.html
index 0175b4227..690164c41 100644
--- a/_modules/geomstats/learning/geometric_median.html
+++ b/_modules/geomstats/learning/geometric_median.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/incremental_frechet_mean.html b/_modules/geomstats/learning/incremental_frechet_mean.html
index ff4da2c4a..ba039f39a 100644
--- a/_modules/geomstats/learning/incremental_frechet_mean.html
+++ b/_modules/geomstats/learning/incremental_frechet_mean.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -429,6 +429,7 @@
                   
   
Source code for geomstats.learning.incremental_frechet_mean
 """Incremental frechet mean estimator."""
+
 from sklearn.base import BaseEstimator
 
 
diff --git a/_modules/geomstats/learning/kalman_filter.html b/_modules/geomstats/learning/kalman_filter.html
index 7b863752a..afa7df595 100644
--- a/_modules/geomstats/learning/kalman_filter.html
+++ b/_modules/geomstats/learning/kalman_filter.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/kernel_density_estimation_classifier.html b/_modules/geomstats/learning/kernel_density_estimation_classifier.html
index 6c2ebcf95..4c2eca12e 100644
--- a/_modules/geomstats/learning/kernel_density_estimation_classifier.html
+++ b/_modules/geomstats/learning/kernel_density_estimation_classifier.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/kmeans.html b/_modules/geomstats/learning/kmeans.html
index c9bfbbfb1..3b6b3428b 100644
--- a/_modules/geomstats/learning/kmeans.html
+++ b/_modules/geomstats/learning/kmeans.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/kmedoids.html b/_modules/geomstats/learning/kmedoids.html
index 3f2a6ccbd..686681fc8 100644
--- a/_modules/geomstats/learning/kmedoids.html
+++ b/_modules/geomstats/learning/kmedoids.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/knn.html b/_modules/geomstats/learning/knn.html
index 1d8584c89..8334ecf2b 100644
--- a/_modules/geomstats/learning/knn.html
+++ b/_modules/geomstats/learning/knn.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/mdm.html b/_modules/geomstats/learning/mdm.html
index 313e9235a..e84f90dd3 100644
--- a/_modules/geomstats/learning/mdm.html
+++ b/_modules/geomstats/learning/mdm.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/online_kmeans.html b/_modules/geomstats/learning/online_kmeans.html
index 8073f2c45..eaaf403fc 100644
--- a/_modules/geomstats/learning/online_kmeans.html
+++ b/_modules/geomstats/learning/online_kmeans.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/pca.html b/_modules/geomstats/learning/pca.html
index c1a729b7e..6039eaddd 100644
--- a/_modules/geomstats/learning/pca.html
+++ b/_modules/geomstats/learning/pca.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/preprocessing.html b/_modules/geomstats/learning/preprocessing.html
index f1b595b83..ac88dee42 100644
--- a/_modules/geomstats/learning/preprocessing.html
+++ b/_modules/geomstats/learning/preprocessing.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/radial_kernel_functions.html b/_modules/geomstats/learning/radial_kernel_functions.html
index c1b6e8d8b..437c6bd23 100644
--- a/_modules/geomstats/learning/radial_kernel_functions.html
+++ b/_modules/geomstats/learning/radial_kernel_functions.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/riemannian_mean_shift.html b/_modules/geomstats/learning/riemannian_mean_shift.html
index 8f48a8b06..5abee5a28 100644
--- a/_modules/geomstats/learning/riemannian_mean_shift.html
+++ b/_modules/geomstats/learning/riemannian_mean_shift.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/learning/wrapped_gaussian_process.html b/_modules/geomstats/learning/wrapped_gaussian_process.html
index 2e4d5014d..84d1f0654 100644
--- a/_modules/geomstats/learning/wrapped_gaussian_process.html
+++ b/_modules/geomstats/learning/wrapped_gaussian_process.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/numerics/bvp.html b/_modules/geomstats/numerics/bvp.html
index 5d5e2fe72..1fd276ad3 100644
--- a/_modules/geomstats/numerics/bvp.html
+++ b/_modules/geomstats/numerics/bvp.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/numerics/geodesic.html b/_modules/geomstats/numerics/geodesic.html
index e8dc59e0e..bbdceb564 100644
--- a/_modules/geomstats/numerics/geodesic.html
+++ b/_modules/geomstats/numerics/geodesic.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/numerics/ivp.html b/_modules/geomstats/numerics/ivp.html
index f4e71f7dc..914fc4211 100644
--- a/_modules/geomstats/numerics/ivp.html
+++ b/_modules/geomstats/numerics/ivp.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/numerics/optimizers.html b/_modules/geomstats/numerics/optimizers.html
index c570b4b61..82f247fbc 100644
--- a/_modules/geomstats/numerics/optimizers.html
+++ b/_modules/geomstats/numerics/optimizers.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/test/data.html b/_modules/geomstats/test/data.html
index 1fa8d601e..1dcb39602 100644
--- a/_modules/geomstats/test/data.html
+++ b/_modules/geomstats/test/data.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/test/random.html b/_modules/geomstats/test/random.html
index ea3b22ce2..03b9ef632 100644
--- a/_modules/geomstats/test/random.html
+++ b/_modules/geomstats/test/random.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/test/utils.html b/_modules/geomstats/test/utils.html
index e51ea778e..22d8b43cc 100644
--- a/_modules/geomstats/test/utils.html
+++ b/_modules/geomstats/test/utils.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/test/vectorization.html b/_modules/geomstats/test/vectorization.html
index 76c1ab675..d5e7ca2a7 100644
--- a/_modules/geomstats/test/vectorization.html
+++ b/_modules/geomstats/test/vectorization.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/test_cases/learning/exponential_barycenter.html b/_modules/geomstats/test_cases/learning/exponential_barycenter.html
index 83894d173..2db6d4a56 100644
--- a/_modules/geomstats/test_cases/learning/exponential_barycenter.html
+++ b/_modules/geomstats/test_cases/learning/exponential_barycenter.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/vectorization.html b/_modules/geomstats/vectorization.html
index 0d877bea5..dfd56e846 100644
--- a/_modules/geomstats/vectorization.html
+++ b/_modules/geomstats/vectorization.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/visualization.html b/_modules/geomstats/visualization.html
index ebb06baad..1fbea6c1f 100644
--- a/_modules/geomstats/visualization.html
+++ b/_modules/geomstats/visualization.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/visualization/hyperbolic.html b/_modules/geomstats/visualization/hyperbolic.html
index ee35735ce..fd672e60a 100644
--- a/_modules/geomstats/visualization/hyperbolic.html
+++ b/_modules/geomstats/visualization/hyperbolic.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -432,6 +432,7 @@
                   
   
Source code for geomstats.visualization.hyperbolic
 """Visualization for Geometric Statistics."""
+
 import matplotlib.pyplot as plt
 from mpl_toolkits.mplot3d import Axes3D  # NOQA
 
diff --git a/_modules/geomstats/visualization/hypersphere.html b/_modules/geomstats/visualization/hypersphere.html
index cb1754e00..e1f6b6cf6 100644
--- a/_modules/geomstats/visualization/hypersphere.html
+++ b/_modules/geomstats/visualization/hypersphere.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/geomstats/visualization/poincare_polydisk.html b/_modules/geomstats/visualization/poincare_polydisk.html
index d5ebe62c8..0d5793e76 100644
--- a/_modules/geomstats/visualization/poincare_polydisk.html
+++ b/_modules/geomstats/visualization/poincare_polydisk.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -432,6 +432,7 @@
                   
   
Source code for geomstats.visualization.poincare_polydisk
 """Visualization for Geometric Statistics."""
+
 import matplotlib.pyplot as plt
 from mpl_toolkits.mplot3d import Axes3D  # NOQA
 
diff --git a/_modules/geomstats/visualization/pre_shape.html b/_modules/geomstats/visualization/pre_shape.html
index 22a94709e..d7303f2a9 100644
--- a/_modules/geomstats/visualization/pre_shape.html
+++ b/_modules/geomstats/visualization/pre_shape.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -432,6 +432,7 @@
                   
   
Source code for geomstats.visualization.pre_shape
 """Visualization for Geometric Statistics."""
+
 import matplotlib.pyplot as plt
 from mpl_toolkits.mplot3d import Axes3D  # NOQA
 
diff --git a/_modules/geomstats/visualization/spd_matrices.html b/_modules/geomstats/visualization/spd_matrices.html
index d83de6054..69d11f121 100644
--- a/_modules/geomstats/visualization/spd_matrices.html
+++ b/_modules/geomstats/visualization/spd_matrices.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -432,6 +432,7 @@
                   
   
Source code for geomstats.visualization.spd_matrices
 """Visualization for Geometric Statistics."""
+
 import matplotlib.pyplot as plt
 from mpl_toolkits.mplot3d import Axes3D  # NOQA
 
diff --git a/_modules/geomstats/visualization/special_euclidean.html b/_modules/geomstats/visualization/special_euclidean.html
index f5fc49aec..6059ea4ab 100644
--- a/_modules/geomstats/visualization/special_euclidean.html
+++ b/_modules/geomstats/visualization/special_euclidean.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
@@ -432,6 +432,7 @@
                   
   
Source code for geomstats.visualization.special_euclidean
 """Visualization for Geometric Statistics."""
+
 import logging
 
 import matplotlib.pyplot as plt
diff --git a/_modules/geomstats/visualization/special_orthogonal.html b/_modules/geomstats/visualization/special_orthogonal.html
index 37bc18a86..c606c82e0 100644
--- a/_modules/geomstats/visualization/special_orthogonal.html
+++ b/_modules/geomstats/visualization/special_orthogonal.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/_modules/index.html b/_modules/index.html
index 19cf4d48a..5f5fb5773 100644
--- a/_modules/index.html
+++ b/_modules/index.html
@@ -45,7 +45,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.datasets.html b/api/geomstats.datasets.html
index 6cfea8a95..99a3b2857 100644
--- a/api/geomstats.datasets.html
+++ b/api/geomstats.datasets.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.distributions.html b/api/geomstats.distributions.html
index 830667771..f9b9d5b26 100644
--- a/api/geomstats.distributions.html
+++ b/api/geomstats.distributions.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.geometry.html b/api/geomstats.geometry.html
index f45e8dea7..da1d044e3 100644
--- a/api/geomstats.geometry.html
+++ b/api/geomstats.geometry.html
@@ -50,7 +50,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.geometry.stratified.html b/api/geomstats.geometry.stratified.html
index d69d020ff..e5543d464 100644
--- a/api/geomstats.geometry.stratified.html
+++ b/api/geomstats.geometry.stratified.html
@@ -50,7 +50,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.html b/api/geomstats.html
index 76ff433a1..b6c7e4597 100644
--- a/api/geomstats.html
+++ b/api/geomstats.html
@@ -50,7 +50,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.information_geometry.html b/api/geomstats.information_geometry.html
index f0b1e67de..d2cb3b030 100644
--- a/api/geomstats.information_geometry.html
+++ b/api/geomstats.information_geometry.html
@@ -50,7 +50,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.learning.html b/api/geomstats.learning.html
index 86966bb55..2b198a120 100644
--- a/api/geomstats.learning.html
+++ b/api/geomstats.learning.html
@@ -50,7 +50,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.numerics.html b/api/geomstats.numerics.html
index a1c564ce8..045c00e6b 100644
--- a/api/geomstats.numerics.html
+++ b/api/geomstats.numerics.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.test.html b/api/geomstats.test.html
index d29f42471..eb5adb722 100644
--- a/api/geomstats.test.html
+++ b/api/geomstats.test.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
@@ -496,17 +496,17 @@ 
geomstats.test.conf module
 

-N_RANDOM_POINTS = [1, 3]#

+N_RANDOM_POINTS = [1, 2]#
 

 
 

 

-N_SHAPE_POINTS = [1, 2]#

+N_SHAPE_POINTS = [1, 4]#
 

 
 

 

-N_TIME_POINTS = [1, 2]#

+N_TIME_POINTS = [1, 4]#
 

 
 

diff --git a/api/geomstats.test_cases.backend.html b/api/geomstats.test_cases.backend.html
index 144e7230f..791d80ba0 100644
--- a/api/geomstats.test_cases.backend.html
+++ b/api/geomstats.test_cases.backend.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.test_cases.datasets.html b/api/geomstats.test_cases.datasets.html
index 496cc2492..5ec493643 100644
--- a/api/geomstats.test_cases.datasets.html
+++ b/api/geomstats.test_cases.datasets.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.test_cases.distributions.html b/api/geomstats.test_cases.distributions.html
index 7c4b335e0..f38112117 100644
--- a/api/geomstats.test_cases.distributions.html
+++ b/api/geomstats.test_cases.distributions.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.test_cases.geometry.html b/api/geomstats.test_cases.geometry.html
index f085aa075..4f6141a90 100644
--- a/api/geomstats.test_cases.geometry.html
+++ b/api/geomstats.test_cases.geometry.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.test_cases.geometry.stratified.html b/api/geomstats.test_cases.geometry.stratified.html
index ba069d9d5..1e49062ca 100644
--- a/api/geomstats.test_cases.geometry.stratified.html
+++ b/api/geomstats.test_cases.geometry.stratified.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.test_cases.html b/api/geomstats.test_cases.html
index 4b1e032fa..9dc18548e 100644
--- a/api/geomstats.test_cases.html
+++ b/api/geomstats.test_cases.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.test_cases.information_geometry.html b/api/geomstats.test_cases.information_geometry.html
index 6509f280f..646b6970e 100644
--- a/api/geomstats.test_cases.information_geometry.html
+++ b/api/geomstats.test_cases.information_geometry.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.test_cases.learning.html b/api/geomstats.test_cases.learning.html
index 929f8d2c7..c64166e11 100644
--- a/api/geomstats.test_cases.learning.html
+++ b/api/geomstats.test_cases.learning.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.test_cases.numerics.html b/api/geomstats.test_cases.numerics.html
index 1ebb81b20..91051df0f 100644
--- a/api/geomstats.test_cases.numerics.html
+++ b/api/geomstats.test_cases.numerics.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/geomstats.visualization.html b/api/geomstats.visualization.html
index 4ba02dfb6..435536579 100644
--- a/api/geomstats.visualization.html
+++ b/api/geomstats.visualization.html
@@ -50,7 +50,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/index.html b/api/index.html
index b5c91eaee..6c91aab8a 100644
--- a/api/index.html
+++ b/api/index.html
@@ -48,7 +48,7 @@
     
   
   
-    
+    
   
   
   
diff --git a/api/modules.html b/api/modules.html
index 992491a4d..795d5361f 100644
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What is the motivation for analyzing data on manifolds?#

 
Many data sets lie on a manifold. The definition of manifold will be introduced later, but for now we can think of a manifold as being something like a surface, such as the surface (2-dimensional sphere) in the figure below.

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Analyzing data that lie on manifolds is often possible without Riemannian Geometry, but choosing to analyze data on manifolds is advantageous for three reasons:

 
    
 
  1. Analyzing data on the manifold it lies on allows you to reduce the degrees of freedom of the system, which makes computations less complicated and more intuitive and interpretable.
    2. 
@@ -492,32 +492,32 @@ 
    1) Analyzing data on a manifold reduces the degrees of freedom of the system
 
    The number of \(\textbf{Degrees of Freedom}\) a system has is equivalent to the number of variables needed to describe the system completely. For example, an object moving freely in three dimensions will require three variables to describe it completely \((x,y,z)\); or \((\phi,\theta, r)\) in spherical coordinates- see Figure (a), below. However, if you know that the point lies on the surface of a sphere of known radius, then you would only need to use two variables,
 \((\theta,\phi)\), to specify its position- see Figure (b), below.
    
 
    Knowing that the point lies on the surface of a manifold (a sphere) allows us to use fewer variables to record its location, which is computationally more efficient in terms of memory requirements, and less mentally taxing (if you are solving a problem on paper).
    
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    Reducing the number of variables is one of the major motivations behind using manifolds to analyze data. Of course, it is sometimes possible to analyze data without manifolds and Riemannian Geometry, but this will be more complicated and computationally intensive.
    
 

 

 
2) Knowing the manifold that a data set belongs to may give you a deeper understanding of your data’s evolution or “why your data looks/evolves the way it does”#

 
Objects travelling along a manifold often follow geodesics on that manifold. When the manifold is equipped with a so-called “Riemannian metric” (which will be introduced in a later notebook), we can define geodesics on the manifold. The geodesic is the path of shortest distance that a particle can travel in the space that it is in. For example, geodesics in 2D and 3D space are straight lines are the shortest way to get from one point to another. The figure below shows paths between two points in
 cartesian space. One path (\(\gamma\)) is a geodesic in cartesian coordinates, and the other path (\(\gamma^{'}\)) is not a geodesic. Figure (a) shows this in 2D, and Figure (b) shows this in 3D.

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However, when an object lies in a higher dimensional curved space, its geodesic will not follow a straight line. For example, if an object is constrained to move along the surface of a sphere, the shortest path between points is not a straight line, but rather, a curve. A straight line in 3D space would not lie on the surface of a sphere, so it cannot be the shortest path along the sphere. The figure below shows \(\gamma\), which is a geodesic on the sphere, along with \(\gamma^{'}\),
 which is not a geodesic on the sphere.

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If you did not know that the object was moving along the surface of the sphere, you would wonder why it is taking such an “irratic” path instead of just going straight. The motion of the particles in your system might seem random because you do not understand the space they are moving in. However, if you learn more about the space they are moving in (the surface of a sphere), you would realize that the particles are following very reasonable and predictable paths along geodesics, and this would
 give you \(\textit{not only}\) a better understanding of how particles have moved in the past but \(\textit{also}\) predictive power to determine how particles will move in the future.

 

 

 
3) Knowing the manifold a data set lives in will help you give you better predictive power and extract the “signal” from a noisy data set or a data set with very few datapoints.#

 
Let’s begin with a simpler example in cartesian coordinates. Let’s say that you are measuring the position of a car moving at constant velocity, but you dropped your position-measuring-device after only one measurement, and your data looks like this:

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How can you get any information from this? It would be very difficult to get information from this if you don’t have an exact model for what the data \(\textit{should}\) look like. Luckily, it is well known that objects moving with constant velocity \(v\) can be modelled by the equation \(x_f = x_i + v\Delta t\), so even though you do not have many data points, you will still be able to accurately predict your car’s future position. However, if you did not know that your data should
 follow a straight line such as in Figure (c) or Figure (d), or you did not know the car’s velocity v such as in Figure (b), then you might try to fit the data using incorrect models, and your (incorrect) model would not be able to accurately predict the car’s future position.

 
In the following figure, the green point in each graph shows the single measured point (the car’s only measured position), and the purple points show future points (the car’s future positions). Note that we are only able to accurately predict future points with a correct model.

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Similarly, knowing the exact manifold your data lies on can help you analyze your current data points and predict future data. For example, let’s say that your data lives on a 2-dimensional sphere, as shown below.

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If you did not know that your data live on the surface of a sphere or you did not, then you might try to fit your data with a line (such as in Figure (b) below) instead of using Geomstats to analyze your data, which appears to follow a geodesic curve on a sphere (shown in Figure (a) below).

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\(\textbf{A note about Geomstat's current capabilities:}\)

 
Our current version of Geomstats cannot tell you \(\textit{which}\) manifold your data lives on, and it also cannot perform “manifold fitting” (adjusting the general parameters of a manifold to fit your data set). For example, Geomstats cannot tell you if your data lives on the surface of a sphere, and if you \(\textit{know}\) that your data falls on the surface of a sphere, Geomstats cannot tell you what the radius of that sphere is.

 
However, if you \(\textit{know}\) the exact manifold that your data lives on (e.g. you know that your data lives on a sphere of radius R=3), then you can \(\textit{use}\) Geomstats to unlock the predictive power that differential geometry can provide you.

@@ -540,7 +540,7 @@ 
What will you learn in these tutorials?

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In the next notebook, we will discuss the Manifold class.

 

 
diff --git a/notebooks/01_foundations__manifolds.html b/notebooks/01_foundations__manifolds.html
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1. Introduction

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9fc4b3bd6dbc49f3a8b027362a999872

 
 

 
2. What is a Manifold?#

@@ -532,8 +532,8 @@ 
2.2 Imprecise “Layman’s Terms” Definition:\(f: (x_1,...,x_d) \to f(x_1,...,x_d)\). This local graph cannot be applied globally (to the whole manifold) because a function \(f\) must have only one output per set of inputs, and if we were to try to describe the whole manifold with such a graph, then many manifolds would have more than one output for one set of inputs.

 
 
The figure below provides a visualization of each of these definitions. The first figure is a visualization of “local parametrization”, the second is a visualization of “local implicit function” and the third is a visualization of “local graph”:

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Each one of these definitions of a manifold is important because each one gives us an additional way to describe a manifold. This is useful because some manifolds are much easier described (or computationally implemented) by one of these definitions than the others. For example, any three of these definitions can be used to describe a hypersphere, but a hypersphere is most easily implemented using definition (2).

 

 

@@ -556,18 +556,18 @@ 
2.3 Hypersphere example:#

 
A hypersphere is one type of manifold, but there are many other types of manifolds that are commonly used and seen in nature. We will show a few examples here to help you build intuition about what a manifold is and what a manifold can look like.

 
For example, a torus (the surface of a donut) is a manifold.

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A klein bottle (the surface of the shape shown below) is also a manifold.

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Perhaps a more familiar example for those in physics: Minkowski space (or Minkowski spacetime) is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold (where x, y, z, and time are the dimensions of the space). This manifold is theorized to take one of three forms. A 4-dimensional manifold of: (a) flat curvature, (b) positive curvature, or (c) negative curvature.

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3. Why do we care about manifolds?#

 
\(\textbf{Manifolds are important because data in nature "naturally falls on manifolds"}\), and as we discussed in the introduction section, knowing the manifold that a data set belongs to may give you more predictive power and a better understanding of the data’s evolution.

 
\(\textbf{What does it mean for data to "naturally fall on a manifold"?}\) In nature, data are often subject to constraints, and these constraints force the data to lie on manifolds. For example, consider the position of cities on the earth.

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The cities are subject to the following constraints: 1) they cannot fly above the surface of the earth because gravity holds them down and 2) they cannot sink down into the earth because the surface of the earth holds them up. Therefore, they are constrained to move (or not move) on the surface of a 3-dimensional sphere. This space that they are confined to exist in should sound familiar– it is the space of a 2-dimensional hypersphere!

 
Other data falls on manifolds in similar ways: the data is subject to certain constraints, which forces it to fall on a manifold.

 

@@ -580,13 +580,13 @@ 
4. What does the #

 
Here we will provide a brief reminder of tangent vectors as a way to lay the groundwork for discussing tangent spaces.

 
A \(\textbf{tangent vector}\) is a vector that is “tangent” to a curve or surface at a given point. When a vector is “tangent” to a curve, this means that the vector has the same slope as the curve does at that point.

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4.2 Tangent Spaces#

 
Now we will introduce the concept of a “tangent space”. The tangent space at a certain point on a manifold is comprised of all of the possible tangent vectors that exist at that point. For example, if you are considering the tangent space on a curve, then the only possible tangent vectors are tangent vectors that point forward and backward along the line (a). However, if you are considering the tangent space at a point on a surface, then the tangent vectors can point forward, backward, left,
 right, and everywhere in between, and the set of all tangent vectors forms a plane (b).

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Thus, the tangent space of a 1-dimensional manifold (curve) is also one dimensional, and the tangent space of a 2-dimensional manifold (a 2-dimensional surface) is also 2-dimensional.

 
Similarly, for every n-dimensional manifold, there exists an n-dimensional tangent space at each point on the manifold, and the tangent space is comprised of all possible tangent vectors on that manifold.

 

@@ -594,7 +594,7 @@ 
4.2 Tangent Spaces
 
5. How is the Manifold class structured?#

 
The hierarchical structure of the classes inheriting from the Manifold parent class is as follows (this Figure is a courtesy of Nicolas Guigui):

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As discussed in the previous section, one of the primary purposes of the Manifold class is to hold information about various types of manifolds. Rules that are universally true for all manifolds are implemented in methods in the parent class Manifold. Rules that are true for some types of manifolds are implemented in the subclasses of Manifold: LevelSet, VectorSpaceOpenSet, FiberBundle, ProductManifold, VectorSpace, MatrixLieAlgebra, and MatrixLieGroup.
 Specific types of manifolds are described in methods within these subclasses.

 
In this notebook, we will focus on describing the subclasses pertinent to the geometry module of geomstats: LevelSet, VectorSpaceOpenSet, ProductManifold and VectorSpace.

@@ -912,9 +912,9 @@ 
5.2 VectorSpace
 

 
5.2.1 What is an Open Set?#

 
Intuitively, an open set is a group of numbers that does not include points on the boundary of whatever they are describing. For example, if you were to take the set of all points between a and b but \(\textbf{not}\) include the values a and b, this would be an open set (shown in figure a). If you were to take the set of all points between a and b and include the values a and b, this would be a closed set (shown in figure b)

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The above examples showed open and closed sets in one dimension. Similarly, in two dimensions, open sets can be defined as sets which do not contain their boundaries. For example, the inside of the sphere, i.e. the ball without its boundary, is a manifold that is an open set. The image below shows an example of an open set (a) and a closed set (b) in two dimensions.

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5.2.2 What Methods are Implemented in VectorSpaceOpenSet?#

@@ -1057,7 +1057,7 @@ 
5.3.1 What is a Level Set?\(\textbf{Level Sets}\). A level set is the set of values \(x\) for which a function f(x) is equal to a given constant. In other words, a level set is a set of curves for which the function describing a manifold is constant along that curve.

 
In the same way that VectorSpaceOpenSet is an implementation of the first definition of a manifold (Local Parametrization), LevelSet is an implementation of the second definition of a manifold (Local Implicit Function). a level set is a set of points for which the function \(f\) takes the exact same value. This value is called the “level”, and does not need to be a scalar, it could also be a vector.

 
For example, consider a hypersphere in three dimensional space. Each of the concentric spheres is a 2-dimensional manifold, each corresponding to a different level \((r1, r2,r3)\).

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You can see here that the subclass Hypersphere is indeed implemented as a LevelSet.

 

 

diff --git a/notebooks/02_foundations__connection_riemannian_metric.html b/notebooks/02_foundations__connection_riemannian_metric.html
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  • Discuss the RiemannianMetric subclass of the Connection Class
    • 
 
 
    This notebook provides an overview of the \(\textbf{Connection}\) class in Geomstats. Intuitively: a connection is a mathematical operator that tells you how much a vector will change when you move it along a manifold in the direction of another vector. The symbol for connection is \(\nabla\), and \(\nabla_{\vec{u}} \vec{v}\) denotes how much vector \(\vec{v}\) would change if you moved it an infinitesimal distance in the direction of vector \(\vec{u}\) (see figure below).
    
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    1. What is a Connection?#
    
 
    We will start by defining the connection \(\textit{mathematically}\), and then we will define the connection again in a more \(\textit{conceptual and intuitive}\) way with examples.
    
@@ -501,7 +501,7 @@ 
    1.1 General Mathematical Definition of the Connection“\(\nabla:\)” - this part is saying that it is the connection (denoted \(\nabla\)) that is performing the transformation.
    
 
    “\(T M\)” - this symbolizes the \(\textbf{tangent bundle}\) of the manifold \(M\). A tangent bundle is the collection of all tangent spaces on the manifold \(M\). A tangent bundle is itself also a manifold.
    
 
    “\(\Gamma(T M)\)” - \(\Gamma(T M)\) denotes the space of vector fields. A single element of \(\Gamma(T M)\) is a single vector field on the manifold \(M\). A vector field is comprised of a set of vectors on the manifold, where there is a vector at each point, and that vector is an element of the tangent space at that point (shown below).
    
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    Putting all of this together, \(\nabla: \Gamma (T M) \times \Gamma (T M) \to \Gamma (T M)\) is simply saying that the connection is a transformation which takes two arguments, both of which are vector fields (comprised of tangent vectors), and the result of the transformation is another vector field (comprised of tangent vectors).
    
 
    \(\textbf{2. Now, let's consider the conditions that the connection must satisfy.}\) 1. (linearity of the \(1^{st}\) argument): \(\nabla_{f X} Y = f \nabla_{X} Y\)
    
 
      
@@ -513,15 +513,15 @@ 
      1.1 General Mathematical Definition of the Connection
 
      1.2 What does the Connection actually do? – a more friendly description of the connection, with an example#
      
 
      Consider a single vector \(\vec{a_p}\), tangent to the manifold at point \(p\) (shown in the figure below).
      
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      Now, suppose you want to see what vector \(\vec{a_p}\) looks like when it is translated to a different point on the manifold. You may be asking why we are placing importance on this question– because for a vector in cartesian coordinates, this question would be trivial. If you translated vector \(\vec{a_p}\) to a different point in cartesian coordinates, the vector itself would not change (as shown in the figure below).
      
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      A spatially translated vector in cartesian coordinates does not change because the basis vectors in cartesian coordinates are translationally invariant, meaning all points in cartesian coordinates have the same set of basis vectors.
      
 
      This is not, however, necessarily true for points on a manifold– if you move a tangent vector from one point to another (i.e. from one tangent space to another), the vector will not necessarily look the same after this transformation. (The figure below shows that when \(\vec{a_p}\) is translated to a different point on the manifold, it could be pointing in a direction different from its original direction.)
      
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      This is why the connection is such an important concept as we analyze data on manifolds. The connection helps us quantify how much a vector will change when we move it from one point to another on a manifold.
      
 
      More specifically, the connection is able to tell us how one vector will change if it is moved in the direction of another vector. For example, if vectors \(\vec{a}\) and \(\vec{b}\) exist in the same tangent plane, then the connection \(\nabla_{\vec{b}} \vec{a}\) tells us how much the vector \(\vec{a}\) will change if it is moved an infinitesimally small distance in the direction of \(\vec{b}\) (shown in the figure below).
      
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@@ -547,7 +547,7 @@ 
    2.1 Parallel Vector Fields\(\textbf{X}\) infinitesimally along the curve \(\gamma\) in the direction of \(\gamma\)’s tangent vector \(\dot{\gamma}\), the vector field \(\textbf{X}\) would not change. Note that this means that in differential geometry, we can only define the term “parallel” when we define it \(\textit{along a curve}\). The figure below shows that at each point \(t\) along the curve \(\gamma\), the vector
 field satisfies \(\nabla_{\dot{\gamma}(t)} X(t)=0\), which means that the vector field shown in pink is a parallel vector field.
    
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    2.2 Parallel Transport#
    
@@ -555,7 +555,7 @@ 
    2.2 Parallel Transport\(T_{\gamma(a)}M\)) to another tangent space (\(T_{\gamma(s)}M\)).
    
 
    Conceptually, this is exactly like solving a differential equation: we can find a “solution” (in this case: the transported vector) to the equation \(\nabla_{\dot \gamma(t)} X(\gamma(t)) = 0\) if we are given initial conditions (in this case: an initial point and tangent vector). Note that \(X(\gamma(t))\) in the differential equation simply indicates that the vector field \(X\) is parallel along the curve \(\gamma(t)\). In other words, the parallel transport calculates the
 parallel vector field along the curve \(\gamma(t)\) and then selects the element of the vector field that lies at the final point \(\gamma(s)\). The figure below shows this process.
    
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    Note that the connection was able to tell us how a vector would change infinitesimally if we move a vector in the direction of another vector, but here, we are able to use parallel transport to calculate how a vector will change if it is translated a \(\textbf{non-infinitesimal}\) distance.
    
 
    
 
@@ -577,7 +577,7 @@ 
    3.3 Building Intuition with Physics\(\textit{four-dimensional}\) “fabric” that makes up the universe, and it is a manifold. Spacetime curves around masses (like planets, stars, black holes), which means that in those areas, we cannot approximate the space as being Euclidean. It
 is in these spaces that we must use geodesics to predict the paths that objects will follow– because free particles (and light) follow geodesics in space.
    
 
    In fact, telescopes have confirmed that light bends around very massive objects like black holes (shown below).
    
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    This is just one example of why geodesics are crucial to understanding real-world data. Other examples exist in areas like biophysics where the trajectory of cells along a manifold may follow geodesics.
    
 
 
@@ -590,7 +590,7 @@ 
    4. Description of Connection in Local Coordinates (using Christoffel Symbols
 
    
 
    4.1 Math Notation: \(\mathbf{ \partial_i}\) Basis Vector#
    
 
    \({\partial_i}\) is equivalent to \(\vec{e}_i\). For example, in three dimensions, one could express the basis vectors \(\hat{x}, \hat{y}, \hat{z}\) as either \(\vec{e}_x, \vec{e}_y, \vec{e}_z\) or as \(\partial_x, \partial_y, \partial_z\).
    
-
    25ecf908a87742138531b2dcaed836e2
    
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    6727582a2b1c499d84e83588d76a496a
    
 
    Note/practicality: basis vectors on a manifold may differ at each point, which means that basis vectors are actually dependent on the tangent space \(T_pM\) to which they belong. Thus, we “should” be writing \(\vec{e}_i(p)\), but in practice, the point \(p\) is omitted.
    
 
    
 
    
@@ -629,27 +629,27 @@ 
    5. Additional Important Mathematical concepts utilized by 
 
    Exponentials on Manifolds#
    
 
    In Euclidean space, \(\textit{addition}\) is a tool which takes two points \(p_1, p_2\), “adds” them, and generates a third, larger point \(p_3\) (as shown in the figure below). Addition gives us a way to “move forward” in Euclidean space.
    
-
    a80b27c87ddf4b02b6cb0f7fdd063bc6
    
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    1f24bcc49599442bbdd63ee8d38d336c
    
 
    On manifolds, the \(\textit{exponential}\) provides a tool which “takes the exponential of the tangent vector at point \(p\)” to generate a third point on the manifold. The exponential does this by 1) identifying the unique geodesic \(\gamma\) that goes through \(p\) and \(v_p\), 2) identifying the “length” \(l\) of the tangent vector \(v_p\), and 3) calculating another point \(p'\) along \(\gamma(t)\) that is a “distance” \(l\) from the initial point
 \(p\) (see figure below). Note again that the notion of “length” and “distance” is different on a manifold than it is in Euclidean space and that quantifying length is not something that we will be able to do without specifying a metric (discussed in RiemannianMetric section).
    
-
    7f84f8d613e94b09b6ffc74f0510f647
    
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    The exponential is often described as being the “differential geometry version” of addition.
    
 
    
 
    
 
    Logarithms on Manifolds#
    
 
    In Euclidean space, \(\textit{subtraction}\) is an operation which allows us to take the third point \(p_3\) and one of the initial points \(p_1\) and extract the other initial point \(p_2\) (as shown in the figure below).
    
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    Similarly, the \(\textit{logarithm}\) allows us to take the final point \(p'\) and the initial point \(p\) to extract the tangent vector \(v_p\) at the initial point. The logarithm is able to do this by 1) identifying the unique geodesic \(\gamma\) that connects the two points 2) calculating the “length” of that geodesic 3) generating the unique tangent vector at \(p\), with a “length” equal to that of the geodesic (shown in the figure below). Again, remember that
 “length” is not something that we can quantify without specifying a metric, which we will not be able to do until the RiemannianMetric section.
    
 
    A key point here is that if you know a point and a tangent vector at that point, you can calculate a unique geodesic that goes through that point, and similarly, if you know the point and geodesic, you should be able to extract the unique tangent vector that produced that geodesic.
    
-
    e489c7978259413091b1c0377ff4a28d
    
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    The logarithm is the inverse map of the exponential and is often described as being the “differential geometry version” of subtraction.
    
 
    
 
 
    
 
    6. Geomstats Documentation: The Connection Class#
    
 
    The Connection class in geomstats has four methods and one subclass, whose structure is shown below:
    
-
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    8f0004ffa5b24f5b9d2ec1384fd6a069
    
 
    The methods of the Connection class also equip the RiemannianMetric subclass with the tools it needs to perform calculations on manifolds (we will discuss the RiemannianMetric class later in this notebook).
    
 
    We will first print the key methods of the Connection class, and then at the end we will print the entire connection class for anyone who is interested.
    
 
    
@@ -1701,7 +1701,7 @@ 
    7.2 How is the Riemannian Metric Connected with the Connection?
 
    7.3 RiemannianMetric in Geomstats#
    
 
    The RiemannianMetric subclass is the class that actually performs calculations on manifolds. This is because we cannot perform “general” calculations like logarithm, exponential, parallel transport, and calculating geodesics without first being equip with a metric. The RiemannianMetric subclass contains a collection of metrics that Geomstats users may use to perform calculations on manifolds. The structure of the RiemannianMetric class is shown below.
    
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     In [7]:
 
    
diff --git a/notebooks/03_practical_methods__data_on_manifolds.html b/notebooks/03_practical_methods__data_on_manifolds.html
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diff --git a/notebooks/04_practical_methods__from_vector_spaces_to_manifolds.html b/notebooks/04_practical_methods__from_vector_spaces_to_manifolds.html
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diff --git a/notebooks/05_practical_methods__simple_machine_learning_on_tangent_spaces.html b/notebooks/05_practical_methods__simple_machine_learning_on_tangent_spaces.html
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diff --git a/notebooks/06_practical_methods__riemannian_frechet_mean_and_tangent_pca.html b/notebooks/06_practical_methods__riemannian_frechet_mean_and_tangent_pca.html
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diff --git a/notebooks/07_practical_methods__riemannian_kmeans.html b/notebooks/07_practical_methods__riemannian_kmeans.html
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diff --git a/notebooks/08_practical_methods__information_geometry.html b/notebooks/08_practical_methods__information_geometry.html
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diff --git a/notebooks/09_practical_methods__implement_your_own_riemannian_geometry.html b/notebooks/09_practical_methods__implement_your_own_riemannian_geometry.html
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diff --git a/notebooks/10_practical_methods__shape_analysis.html b/notebooks/10_practical_methods__shape_analysis.html
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diff --git a/notebooks/11_real_world_applications__cell_shapes_analysis.html b/notebooks/11_real_world_applications__cell_shapes_analysis.html
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    Shape Analysis of Cancer CellsICLR 2021 Computational Geometry and Topology challenge.
    
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    1f1c2b58f31940a2895763fa3c4a29ae
    
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    527aa37a89b042e5860b28de4d4691d5
    
 
    Figure 1: Representative images of the cell lines using fluorescence microscopy, studied in this notebook (Image credit : Ashok Prasad). The cells nuclei (blue), the actin cytoskeleton (green) and the lipid membrane (red) of each cell are stained and colored. We only focus on the cell shape in our analysis.
    
 
    
 
    
@@ -530,7 +530,7 @@ 
    2. Dataset DescriptionWe study a dataset of mouse Osteosarcoma imaged cells (AXCFP2019). The dataset contains two different cancer cell lines : DLM8 and DUNN, respectively representing a more agressive and a less agressive cancer. Among these cells, some have also been treated with different single drugs that perturb the cellular cytoskeleton. Overall, we can label each cell according to their cell line (DLM8 and DUNN), and also if it is a control cell (no treatment), or has been treated
 with one of the following drugs : Jasp (jasplakinolide) and Cytd (cytochalasin D).
    
 
    Each cell comes from a raw image containing a set of cells, which was thresholded to generate binarized images.
    
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    735f1514192f4cf9b692f34faebd8671
    
 
    After binarizing the images, contouring was used to isolate each cell, and to extract their boundaries as a counter-clockwise ordered list of 2D coordinates, which corresponds to the representation of discrete curve in Geomstats. We load these discrete curves into the notebook.
    
 
    
 
     In [2]:
diff --git a/notebooks/12_real_world_applications__emg_sign_classification_in_spd_manifold.html b/notebooks/12_real_world_applications__emg_sign_classification_in_spd_manifold.html
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    Hand gesture classification with EMG data using Riemannian metrics
    
+
    f304da3a75a0416195613ccf9acfb7be
    
 
    
 
    Context#
    
 
    The data are acquired from somOS-interface: an sEMG armband that allows you to interact via bluetooth with an Android smartphone (you can contact Marius Guerard (marius.guerard@gmail.com) or Renaud Renault (renaud.armand.renault@gmail.com) for more info on how to make this kind of armband yourself).
    
diff --git a/notebooks/13_real_world_applications__graph_embedding_and_clustering_in_hyperbolic_space.html b/notebooks/13_real_world_applications__graph_embedding_and_clustering_in_hyperbolic_space.html
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    Parameters and Initialization Some information about the dataset is displayed to provide insight into its
+
    The Zachary karate club network was collected from the members of a university karate club by Wayne Zachary in 1977. Each node represents a member of the club, and each edge represents an undirected relation between two members. An often discussed problem using this dataset is to find the two groups of people into which the karate club split after an argument between two teachers. 4006e7ae8ef9426da09364401d3ce9c4 Some information about the dataset is displayed to provide insight into its
 complexity.
    
 
    
 
     In [3]:
@@ -598,7 +598,7 @@ 
    Parameters and Initialization\(V\) as the set of nodes and \(E \subset V\times V\) the set of edges. The goal of embedding GSD is to provide a faithful and exploitable representation of the graph structure. It is mainly achieved by preserving first-order proximity that enforces nodes sharing edges to be close to each other. It can additionally preserve second-order proximity that enforces two nodes sharing the same context (i.e., nodes that share neighbors but are not necessarily directly connected) to be
 close. Let \(\mathbb{B}^m\) be the Poincaré Ball of dimension \(m\) equipped with the distance function \(d\). The below figure shows geodesics between pairs of points on \(\mathbb{B}^2\). Geodesics are the shortest path between two points. The distance function \(d\) of two points is the length of the geodesic that links them.
    
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    Declaring an instance of the PoincareBall manifold of two dimensions in geomstats is straightforward:
    
 
    
 
     In [4]:
@@ -616,7 +616,7 @@ 
    Loss function.
 \[\mathcal{L} = - \sum_{v_i\in V} \sum_{v_j \in C_i} \bigg[ log(\sigma(-d^2(\phi_i, \phi_j'))) + \sum_{v_k\sim \mathcal{P}_n} log(\sigma(d^2(\phi_i, \phi_k')))  \bigg]\]
    
 
    where \(\sigma(x)=\frac{1}{1+e^{-x}}\) is the sigmoid function and \(\phi_i \in \mathbb{B}^m\) is the embedding of the \(i\)-th node of \(V\), \(C_i\) the nodes in the context of the \(i\)-th node, \(\phi_j'\in \mathbb{B}^m\) the embedding of \(v_j\in C_i\) and \(\mathcal{P}_n\) the negative sampling distribution over \(V\): \(\mathcal{P}_n(v)=\frac{deg(v)^{3/4}}{\sum_{v_i\in V}deg(v_i)^{3/4}}\). Intuitively one can see that to minimizing \(L\),
-the distance between \(v_i\) and \(v_j\) should get smaller, while the one between \(v_i\) and \(v_k\) would get larger. fa34e1d75c154f0388ce236865ff0222
    
+the distance between \(v_i\) and \(v_j\) should get smaller, while the one between \(v_i\) and \(v_k\) would get larger. 9a17ed50de864d4483bb33418dc2d2a0
    
 
    
 
    
 
    Riemannian optimization.#
    
diff --git a/notebooks/14_real_world_applications__hand_poses_analysis_in_kendall_shape_space.html b/notebooks/14_real_world_applications__hand_poses_analysis_in_kendall_shape_space.html
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    Classifying hands poses with Kendall shape spacesIn this tutorial, we show how to use geomstats to perform a shape data analysis. Specifically, we aim to study the difference between two groups of data: - hand poses that correspond to the action “Grab”, - hand poses heads that correspond to the action “Expand”.
    
 
    We wish to investigate if there is a difference in these two groups.
    
 
    The hand poses are represented as the coordinates of 22 joints in 3D:
    
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     In [1]:
 
    
diff --git a/notebooks/15_real_world_applications__optic_nerve_heads_analysis_in_kendall_shape_space.html b/notebooks/15_real_world_applications__optic_nerve_heads_analysis_in_kendall_shape_space.html
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    Computing with shapes of landmarks in Kendall shape spaces
    
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     In [1]:
 
    
diff --git a/notebooks/16_real_world_applications__visualizations_in_kendall_shape_spaces.html b/notebooks/16_real_world_applications__visualizations_in_kendall_shape_spaces.html
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diff --git a/notebooks/17_foundations__stratified_spaces.html b/notebooks/17_foundations__stratified_spaces.html
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    Stratified spaces
 
    Spider#
    
 
    The \(k\)-Spider consists of \(k\) copies of the positive real line \(\mathbb{R}_{\geq 0}\) glued together at the origin. Within geomstats, we defined the following: 1. Spider Point: a point object defining the ray and the value 2. Spider: the space defined by the number of rays 3. Spider Geometry: by chosing a metric on the rays, we can define a metric on the whole space
    
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diff --git a/notebooks/18_real_world_applications__sao_paulo_traffic_optimization.html b/notebooks/18_real_world_applications__sao_paulo_traffic_optimization.html
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    Optimization of Sao Paulo traffic
    
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    ddea9c72ec1747a99acbb71238d5704f
    
 
    Figure 1: Sao Paulo: A city with 180km traffic jams – BBC News
    
 
    
 
     In [1]:
@@ -1247,7 +1247,7 @@ 
    3.1.2. Model\(r\), we note \(T_r\) the congestion time that cars will have to wait in the case the traffic is jammed: \(T_r \rightsquigarrow G(k_r, \nu_r)\), with distribution:
    
 
    
 \[\forall t>0, \, \mathbb{f}(t) = \frac{\nu_r^{k_r}}{\Gamma(k_r)} t^{k_r-1} e^{-\nu_r t}.\]
    
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    As a road \(x_r\) can be represented by two parameters \(k_r\) and \(\nu_r\), we can consider our space of study to be the space of such parameters (i.e. \(\mathbb{(R_+^*)^2}\)).
    
 
    For the following, we denote Gamma distributions’ parameters by \((\kappa_r, \gamma_r)\), where \(\kappa_r\)=\(k_r\) (expected jam size) and \(\gamma_r\)=\(\frac{k_r}{\nu_r}\) is the expected congestion time (mean of the Gamma distribution). The space of study is still \(\mathbb{(R_+^*)^2}\), and we are instantiating it in the next cell.
    
 
    
diff --git a/notebooks/19_practical_methods__aac.html b/notebooks/19_practical_methods__aac.html
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diff --git a/notebooks/20_real_world_applications__graph_space.html b/notebooks/20_real_world_applications__graph_space.html
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"geomstats-test-cases-geometry-stratified-package"]], "geomstats.test_cases.geometry.stratified.point_set module": [[14, "geomstats-test-cases-geometry-stratified-point-set-module"]], "geomstats.test_cases.information_geometry package": [[15, "geomstats-test-cases-information-geometry-package"]], "geomstats.test_cases.information_geometry.base module": [[15, "geomstats-test-cases-information-geometry-base-module"]], "geomstats.test_cases.information_geometry.beta module": [[15, "geomstats-test-cases-information-geometry-beta-module"]], "geomstats.test_cases.information_geometry.binomial module": [[15, "geomstats-test-cases-information-geometry-binomial-module"]], "geomstats.test_cases.information_geometry.dirichlet module": [[15, "geomstats-test-cases-information-geometry-dirichlet-module"]], "geomstats.test_cases.information_geometry.exponential module": [[15, "geomstats-test-cases-information-geometry-exponential-module"]], "geomstats.test_cases.information_geometry.gamma module": [[15, "geomstats-test-cases-information-geometry-gamma-module"]], "geomstats.test_cases.information_geometry.geometric module": [[15, "geomstats-test-cases-information-geometry-geometric-module"]], "geomstats.test_cases.information_geometry.multinomial module": [[15, "geomstats-test-cases-information-geometry-multinomial-module"]], "geomstats.test_cases.information_geometry.normal module": [[15, "geomstats-test-cases-information-geometry-normal-module"]], "geomstats.test_cases.information_geometry.poisson module": [[15, "geomstats-test-cases-information-geometry-poisson-module"]], "geomstats.test_cases.learning package": [[16, "geomstats-test-cases-learning-package"]], "geomstats.test_cases.learning.agglomerative_hierarchical_clustering module": [[16, "geomstats-test-cases-learning-agglomerative-hierarchical-clustering-module"]], "geomstats.test_cases.learning.expectation_maximization module": [[16, "geomstats-test-cases-learning-expectation-maximization-module"]], "geomstats.test_cases.learning.exponential_barycenter module": [[16, "module-geomstats.test_cases.learning.exponential_barycenter"]], "geomstats.test_cases.learning.frechet_mean module": [[16, "geomstats-test-cases-learning-frechet-mean-module"]], "geomstats.test_cases.learning.geodesic_regression module": [[16, "geomstats-test-cases-learning-geodesic-regression-module"]], "geomstats.test_cases.learning.incremental_frechet_mean module": [[16, "geomstats-test-cases-learning-incremental-frechet-mean-module"]], "geomstats.test_cases.learning.kalman_filter module": [[16, "geomstats-test-cases-learning-kalman-filter-module"]], "geomstats.test_cases.learning.kernel_density_estimation_classifier module": [[16, "geomstats-test-cases-learning-kernel-density-estimation-classifier-module"]], "geomstats.test_cases.learning.kmeans module": [[16, "geomstats-test-cases-learning-kmeans-module"]], "geomstats.test_cases.learning.knn module": [[16, "geomstats-test-cases-learning-knn-module"]], "geomstats.test_cases.learning.mdm module": [[16, "geomstats-test-cases-learning-mdm-module"]], "geomstats.test_cases.learning.pca module": [[16, "geomstats-test-cases-learning-pca-module"]], "geomstats.test_cases.learning.preprocessing module": [[16, "geomstats-test-cases-learning-preprocessing-module"]], "geomstats.test_cases.learning.radial_kernel_functions module": [[16, "geomstats-test-cases-learning-radial-kernel-functions-module"]], "geomstats.test_cases.learning.wrapped_gaussian_process module": [[16, "geomstats-test-cases-learning-wrapped-gaussian-process-module"]], "geomstats.test_cases.numerics package": [[17, "geomstats-test-cases-numerics-package"]], "geomstats.test_cases.numerics.geodesic module": [[17, "geomstats-test-cases-numerics-geodesic-module"]], "geomstats.visualization package": [[18, "geomstats-visualization-package"]], "geomstats.visualization.hyperbolic module": [[18, "module-geomstats.visualization.hyperbolic"]], "geomstats.visualization.hypersphere module": [[18, "module-geomstats.visualization.hypersphere"]], "geomstats.visualization.poincare_polydisk module": [[18, "module-geomstats.visualization.poincare_polydisk"]], "geomstats.visualization.pre_shape module": [[18, "module-geomstats.visualization.pre_shape"]], "geomstats.visualization.spd_matrices module": [[18, "module-geomstats.visualization.spd_matrices"]], "geomstats.visualization.special_euclidean module": [[18, "module-geomstats.visualization.special_euclidean"]], "geomstats.visualization.special_orthogonal module": [[18, "module-geomstats.visualization.special_orthogonal"]], "API Reference": [[19, "api-reference"]], "Packages & Modules": [[19, null]], "geomstats": [[20, "geomstats"]], "Contributing Guide": [[21, "contributing-guide"]], "Development Setup": [[21, "development-setup"]], "Source control with Git": [[21, "source-control-with-git"]], "Getting the code": [[21, "getting-the-code"]], "Dependencies and a virtual environment": [[21, "dependencies-and-a-virtual-environment"]], "Backends": [[21, "backends"]], "Run the tests": [[21, "run-the-tests"]], "Build the docs": [[21, "build-the-docs"]], "Folder Structure": [[21, "folder-structure"]], "Testing": [[21, "testing"]], "Test Driven Development": [[21, "test-driven-development"]], "TDD with pytest": [[21, "tdd-with-pytest"]], "Writing tests for geomstats": [[21, "writing-tests-for-geomstats"]], "Running tests": [[21, "running-tests"]], "Workflow of a contribution": [[21, "workflow-of-a-contribution"]], "Create or choose an issue for new contributors": [[21, "create-or-choose-an-issue-for-new-contributors"]], "good first issue tag": [[21, null]], "starter project tag": [[21, null]], "help wanted tag": [[21, null]], "Making changes": [[21, "making-changes"]], "Pull Request Checklist": [[21, "pull-request-checklist"]], "Stalled Pull Requests": [[21, "stalled-pull-requests"]], "Coding Style Guidelines": [[21, "coding-style-guidelines"]], "Documentation": [[21, "documentation"]], "Building the Documentation": [[21, "building-the-documentation"]], "Writing Docstrings": [[21, "writing-docstrings"]], "Intro to Docstrings": [[21, "intro-to-docstrings"]], "The Anatomy of a Docstring": [[21, "the-anatomy-of-a-docstring"]], "Docstring Examples": [[21, "docstring-examples"]], "Code Review Guidelines": [[21, "code-review-guidelines"]], "Reporting bugs and features": [[21, "reporting-bugs-and-features"]], "Issue tracker": [[21, "issue-tracker"]], "Template of a bug/issue report": [[21, "template-of-a-bug-issue-report"]], "Issue Triaging": [[21, "issue-triaging"]], "Reproducing issues": [[21, "reproducing-issues"]], "Commenting on alternative solutions": [[21, "commenting-on-alternative-solutions"]], "Answering questions": [[21, "answering-questions"]], "Labelling and assigning the issue": [[21, "labelling-and-assigning-the-issue"]], "Explanation": [[22, "explanation"]], "Code Examples": [[23, "code-examples"]], "Learn differential geometry": [[23, "learn-differential-geometry"]], "Run learning algorithms": [[23, "run-learning-algorithms"]], "First steps": [[24, "first-steps"]], "Getting Started": [[25, "getting-started"]], "Geomstats governance and decision-making": [[26, "geomstats-governance-and-decision-making"]], "Roles And Responsibilities": [[26, "roles-and-responsibilities"]], "Contributors": [[26, "contributors"]], "Contributor Experience Team": [[26, "contributor-experience-team"]], "Communication team": [[26, "communication-team"]], "Core developers": [[26, "core-developers"]], "Technical Committee": [[26, "technical-committee"]], "Decision Making Process": [[26, "decision-making-process"]], "Conflict of Interest": [[26, "conflict-of-interest"]], "Breach": [[26, "breach"]], "Changing the Governance Documents": [[26, "changing-the-governance-documents"]], "Google Season of Docs: Case Study": [[27, "google-season-of-docs-case-study"]], "Design and Organize Scattered Documentation in Geomstats": [[27, "design-and-organize-scattered-documentation-in-geomstats"]], "Problem Statement": [[27, "problem-statement"]], "Proposal Abstract": [[27, "proposal-abstract"]], "Project Description": [[27, "project-description"]], "Budget": [[27, "budget"]], "Participants": [[27, "participants"]], "Timeline and Results": [[27, "timeline-and-results"]], "Title": [[27, "id4"]], "Metrics": [[27, "metrics"]], "Analysis": [[27, "analysis"]], "Summary": [[27, "summary"]], "Google Season of Docs: Project Page": [[27, "google-season-of-docs-project-page"]], "About the organization": [[27, "about-the-organization"]], "About the project": [[27, "about-the-project"]], "The problem": [[27, "the-problem"]], "The project\u2019s scope": [[27, "the-projects-scope"]], "Measuring project\u2019s success": [[27, "measuring-projects-success"]], "Proposed Budget": [[27, "proposed-budget"]], "Additional information": [[27, "additional-information"]], "Important links for technical writers": [[27, "important-links-for-technical-writers"]], "Hackathons": [[28, "hackathons"]], "Workflow": [[28, "workflow"]], "Past and future hackathons": [[28, "past-and-future-hackathons"]], "Related events": [[28, "related-events"]], "Hackathon at Institut Henri Poincar\u00e9, 17th-21st Oct 2022": [[29, "hackathon-at-institut-henri-poincare-17th-21st-oct-2022"]], "Future changes": [[29, "future-changes"]], "Equip with metric": [[29, "equip-with-metric"]], "Intrinsic vs extrinsic coordinates (manifold)": [[29, "intrinsic-vs-extrinsic-coordinates-manifold"]], "Intrinsic vs extrinsic coordinates (connection)": [[29, "intrinsic-vs-extrinsic-coordinates-connection"]], "No dynamic behavior": [[29, "no-dynamic-behavior"]], "numerics": [[29, "numerics"]], "gs.matrices": [[29, "gs-matrices"]], "Points shape": [[29, "points-shape"]], "New structure": [[29, "new-structure"]], "Geomstats": [[30, "geomstats"]], "Objectives": [[30, "objectives"]], "Design": [[30, "design"]], "Learn More": [[30, "learn-more"]], "What is the motivation for analyzing data on manifolds?": [[31, "What-is-the-motivation-for-analyzing-data-on-manifolds?"]], "1) Analyzing data on a manifold reduces the degrees of freedom of the system, making computations less complicated.": [[31, "1)-Analyzing-data-on-a-manifold-reduces-the-degrees-of-freedom-of-the-system,-making-computations-less-complicated."]], "2) Knowing the manifold that a data set belongs to may give you a deeper understanding of your data\u2019s evolution or \u201cwhy your data looks/evolves the way it does\u201d": [[31, "2)-Knowing-the-manifold-that-a-data-set-belongs-to-may-give-you-a-deeper-understanding-of-your-data's-evolution-or-%22why-your-data-looks/evolves-the-way-it-does%22"]], "3) Knowing the manifold a data set lives in will help you give you better predictive power and extract the \u201csignal\u201d from a noisy data set or a data set with very few datapoints.": [[31, "3)-Knowing-the-manifold-a-data-set-lives-in-will-help-you-give-you-better-predictive-power-and-extract-the-%22signal%22-from-a-noisy-data-set-or-a-data-set-with-very-few-datapoints."]], "What will you learn in these tutorials?": [[31, "What-will-you-learn-in-these-tutorials?"]], "1. Introduction": [[32, "1.-Introduction"]], "2. What is a Manifold?": [[32, "2.-What-is-a-Manifold?"]], "2.1 Precise Mathematical Definition": [[32, "2.1-Precise-Mathematical-Definition"]], "2.2 Imprecise \u201cLayman\u2019s Terms\u201d Definition:": [[32, "2.2-Imprecise-%22Layman's-Terms%22-Definition:"]], "2.3 Hypersphere example:": [[32, "2.3-Hypersphere-example:"]], "2.4 More examples of manifolds:": [[32, "2.4-More-examples-of-manifolds:"]], "3. Why do we care about manifolds?": [[32, "3.-Why-do-we-care-about-manifolds?"]], "4. What does the Manifold class do?": [[32, "4.-What-does-the-Manifold-class-do?"]], "4.1 Tangent Vectors": [[32, "4.1-Tangent-Vectors"]], "4.2 Tangent Spaces": [[32, "4.2-Tangent-Spaces"]], "5. How is the Manifold class structured?": [[32, "5.-How-is-the-Manifold-class-structured?"]], "5.1 The Parent Class: Manifold": [[32, "5.1-The-Parent-Class:-Manifold"]], "5.1.1 Examples of Using Manifold\u2019s Attributes and Methods in the Subclass: Hypersphere": [[32, "5.1.1-Examples-of-Using-Manifold's-Attributes-and-Methods-in-the-Subclass:-Hypersphere"]], "5.1.2 The Full Manifold Class Code": [[32, "5.1.2-The-Full-Manifold-Class-Code"]], "5.2 VectorSpaceOpenSet": [[32, "5.2-VectorSpaceOpenSet"]], "5.2.1 What is an Open Set?": [[32, "5.2.1-What-is-an-Open-Set?"]], "5.2.2 What Methods are Implemented in VectorSpaceOpenSet?": [[32, "5.2.2-What-Methods-are-Implemented-in-VectorSpaceOpenSet?"]], "5.3 LevelSet": [[32, "5.3-LevelSet"]], "5.3.1 What is a Level Set?": [[32, "5.3.1-What-is-a-Level-Set?"]], "5.3.2 LevelSet in Geomstats": [[32, "5.3.2-LevelSet-in-Geomstats"]], "5.4 VectorSpace": [[32, "5.4-VectorSpace"]], "5.5 ProductManifold": [[32, "5.5-ProductManifold"]], "5.5.1 Example": [[32, "5.5.1-Example"]], "Conclusion": [[32, "Conclusion"], [36, "Conclusion"], [49, "Conclusion"]], "1. What is a Connection?": [[33, "1.-What-is-a-Connection?"]], "1.1 General Mathematical Definition of the Connection": [[33, "1.1-General-Mathematical-Definition-of-the-Connection"]], "1.2 What does the Connection actually do? \u2013 a more friendly description of the connection, with an example": [[33, "1.2-What-does-the-Connection-actually-do?----a-more-friendly-description-of-the-connection,-with-an-example"]], "2. Why is the Connection important (big picture)? \u2013 parallel transport, parallel vector fields": [[33, "2.-Why-is-the-Connection-important-(big-picture)?----parallel-transport,-parallel-vector-fields"]], "2.1 Parallel Vector Fields": [[33, "2.1-Parallel-Vector-Fields"]], "2.2 Parallel Transport": [[33, "2.2-Parallel-Transport"]], "3. Defining Geodesics with the Connection": [[33, "3.-Defining-Geodesics-with-the-Connection"]], "3.1 Intuitive Definition of a Geodesic": [[33, "3.1-Intuitive-Definition-of-a-Geodesic"]], "3.2 Defining a geodesic with the connection": [[33, "3.2-Defining-a-geodesic-with-the-connection"]], "3.3 Building Intuition with Physics": [[33, "3.3-Building-Intuition-with-Physics"]], "4. Description of Connection in Local Coordinates (using Christoffel Symbols)": [[33, "4.-Description-of-Connection-in-Local-Coordinates-(using-Christoffel-Symbols)"]], "4.1 Math Notation: \\mathbf{ \\partial_i} Basis Vector": [[33, "4.1-Math-Notation:-\\mathbf{-\\partial_i}-Basis-Vector"]], "4.2 Einstein Notation": [[33, "4.2-Einstein-Notation"]], "4.3 What does a Christoffel Symbol represent?": [[33, "4.3-What-does-a-Christoffel-Symbol-represent?"]], "4.4 The Geodesic Equation in local coordinates": [[33, "4.4-The-Geodesic-Equation-in-local-coordinates"]], "4.5 Why should we care about Christoffel Symbols and this representation of the connection?": [[33, "4.5-Why-should-we-care-about-Christoffel-Symbols-and-this-representation-of-the-connection?"]], "5. Additional Important Mathematical concepts utilized by Connection Class: Logarithm, Exponential": [[33, "5.-Additional-Important-Mathematical-concepts-utilized-by-Connection-Class:-Logarithm,-Exponential"]], "Exponentials on Manifolds": [[33, "Exponentials-on-Manifolds"]], "Logarithms on Manifolds": [[33, "Logarithms-on-Manifolds"]], "6. Geomstats Documentation: The Connection Class": [[33, "6.-Geomstats-Documentation:-The-Connection-Class"]], "6.1 method: parallel_transport()": [[33, "6.1-method:-parallel_transport()"]], "6.2 method: geodesic_equation()": [[33, "6.2-method:-geodesic_equation()"]], "6.3 method: exp() (exponential)": [[33, "6.3-method:-exp()-(exponential)"]], "6.4 method: log() (logarithm)": [[33, "6.4-method:-log()-(logarithm)"]], "6.5 The full Connection class": [[33, "6.5-The-full-Connection-class"]], "7. RiemannianMetric Sub-Class": [[33, "7.-RiemannianMetric-Sub-Class"]], "7.1 Riemannian Metrics": [[33, "7.1-Riemannian-Metrics"]], "7.2 How is the Riemannian Metric Connected with the Connection?": [[33, "7.2-How-is-the-Riemannian-Metric-Connected-with-the-Connection?"]], "7.3 RiemannianMetric in Geomstats": [[33, "7.3-RiemannianMetric-in-Geomstats"]], "8. Conclusion": [[33, "8.-Conclusion"]], "Data on Manifolds": [[34, "Data-on-Manifolds"]], "From data on linear spaces to data on manifolds": [[34, "From-data-on-linear-spaces-to-data-on-manifolds"]], "Examples of data on manifolds": [[34, "Examples-of-data-on-manifolds"]], "World cities: data on the sphere": [[34, "World-cities:-data-on-the-sphere"]], "Pose of objects in pictures: data on the Lie group of 3D rotations": [[34, "Pose-of-objects-in-pictures:-data-on-the-Lie-group-of-3D-rotations"]], "Social networks: data on the hyperbolic space": [[34, "Social-networks:-data-on-the-hyperbolic-space"]], "Brain connectomes: data on the manifold of Symmetric Positive Definite (SPD) matrices": [[34, "Brain-connectomes:-data-on-the-manifold-of-Symmetric-Positive-Definite-(SPD)-matrices"]], "Monkey\u2019s optical nerve heads: Data as landmarks in 3D": [[34, "Monkey's-optical-nerve-heads:-Data-as-landmarks-in-3D"]], "Reference:": [[34, "Reference:"]], "From vector spaces to manifolds": [[35, "From-vector-spaces-to-manifolds"], [35, "id1"]], "From addition to exponential map": [[35, "From-addition-to-exponential-map"]], "From substraction to logarithm map": [[35, "From-substraction-to-logarithm-map"]], "Geodesics": [[35, "Geodesics"], [47, "Geodesics"]], "Geodesics on the hyperbolic plane": [[35, "Geodesics-on-the-hyperbolic-plane"]], "Geodesics on the special euclidean group SE(3)": [[35, "Geodesics-on-the-special-euclidean-group-SE(3)"]], "Learning on Tangent Data": [[36, "Learning-on-Tangent-Data"]], "The Data": [[36, "The-Data"], [43, "The-Data"]], "The Manifold": [[36, "The-Manifold"]], "The Transformer": [[36, "The-Transformer"]], "References": [[36, "References"], [39, "References"], [41, "References"], [44, "References"], [45, "References"], [46, "References"], [47, "References"]], "Fr\u00e9chet Mean and Tangent PCA": [[37, "Fr\u00e9chet-Mean-and-Tangent-PCA"]], "On the sphere": [[37, "On-the-sphere"]], "Generate data on the sphere": [[37, "Generate-data-on-the-sphere"]], "Fr\u00e9chet mean": [[37, "Fr\u00e9chet-mean"], [37, "id1"]], "Tangent PCA (at the Fr\u00e9chet mean)": [[37, "Tangent-PCA-(at-the-Fr\u00e9chet-mean)"], [37, "id2"]], "In the Hyperbolic plane": [[37, "In-the-Hyperbolic-plane"]], "Generate data on the hyperbolic plane": [[37, "Generate-data-on-the-hyperbolic-plane"]], "K-Means clustering on a Riemannian Manifold": [[38, "K-Means-clustering-on-a-Riemannian-Manifold"]], "Data init: Generate clusters randomly on the sphere": [[38, "Data-init:-Generate-clusters-randomly-on-the-sphere"]], "Operate Riemannian K-Means clustering": [[38, "Operate-Riemannian-K-Means-clustering"]], "Information geometry": [[39, "Information-geometry"]], "Introduction": [[39, "Introduction"], [44, "Introduction"]], "Normal distributions": [[39, "Normal-distributions"]], "Beta distributions": [[39, "Beta-distributions"]], "Implement your own Riemannian Geometry": [[40, "Implement-your-own-Riemannian-Geometry"]], "Immersed manifolds: the example of the 2-sphere": [[40, "Immersed-manifolds:-the-example-of-the-2-sphere"]], "Immersed manifolds: the example of a surface defined by its graph": [[40, "Immersed-manifolds:-the-example-of-a-surface-defined-by-its-graph"]], "Pullback via diffeomorphism: Warping of the plane \\mathbb{R}^2": [[40, "Pullback-via-diffeomorphism:-Warping-of-the-plane-\\mathbb{R}^2"]], "Shape analysis of curves with the Square Root Velocity metric": [[41, "Shape-analysis-of-curves-with-the-Square-Root-Velocity-metric"]], "1. A first example in 2D": [[41, "1.-A-first-example-in-2D"]], "Remarks:": [[41, "Remarks:"]], "2. Focus on quotienting out reparametrizations": [[41, "2.-Focus-on-quotienting-out-reparametrizations"]], "2.1. Comparing parametrized curves": [[41, "2.1.-Comparing-parametrized-curves"]], "2.2. Comparing unparametrized curves": [[41, "2.2.-Comparing-unparametrized-curves"]], "3. Example in 3D": [[41, "3.-Example-in-3D"]], "Shape Analysis of Cancer Cells": [[42, "Shape-Analysis-of-Cancer-Cells"]], "1. Introduction and Motivation": [[42, "1.-Introduction-and-Motivation"], [49, "1.-Introduction-and-Motivation"]], "2. Dataset Description": [[42, "2.-Dataset-Description"]], "3. Preprocessing": [[42, "3.-Preprocessing"]], "Interpolation: Encoding Discrete Curves With Same Number of Points": [[42, "Interpolation:-Encoding-Discrete-Curves-With-Same-Number-of-Points"]], "Visualization of Interpolated Dataset of Curves": [[42, "Visualization-of-Interpolated-Dataset-of-Curves"]], "Remove duplicate samples in curves": [[42, "Remove-duplicate-samples-in-curves"]], "Alignment": [[42, "Alignment"]], "4 Data Analysis": [[42, "4-Data-Analysis"]], "Analyze Distances to the \u201cGlobal\u201d Mean Shape": [[42, "Analyze-Distances-to-the-%22Global%22-Mean-Shape"]], "Changing the Metric on the Space of Cell Shapes": [[42, "Changing-the-Metric-on-the-Space-of-Cell-Shapes"]], "Hypothesis Testing on Equality of Means": [[42, "Hypothesis-Testing-on-Equality-of-Means"]], "Visualization of the Mean of each Treatment": [[42, "Visualization-of-the-Mean-of-each-Treatment"]], "Distance of the Cell Shapes to their Own Mean Shape": [[42, "Distance-of-the-Cell-Shapes-to-their-Own-Mean-Shape"]], "Hand gesture classification with EMG data using Riemannian metrics": [[43, "Hand-gesture-classification-with-EMG-data-using-Riemannian-metrics"]], "Context": [[43, "Context"]], "Parameters": [[43, "Parameters"]], "Preprocessing into covariance matrices": [[43, "Preprocessing-into-covariance-matrices"]], "Covariances plot of the euclidean average": [[43, "Covariances-plot-of-the-euclidean-average"]], "Covariances plot of the Frechet Mean of the affine invariant metric": [[43, "Covariances-plot-of-the-Frechet-Mean-of-the-affine-invariant-metric"]], "Sign Classification": [[43, "Sign-Classification"]], "A. Test on the same session and user as Training/Calibration": [[43, "A.-Test-on-the-same-session-and-user-as-Training/Calibration"]], "A.0. Using Logistic Regression on the vectorized Matrix (Euclidean Method)": [[43, "A.0.-Using-Logistic-Regression-on-the-vectorized-Matrix-(Euclidean-Method)"]], "A.1. Using MLP on the vectorized Matrix (Euclidean Method)": [[43, "A.1.-Using-MLP-on-the-vectorized-Matrix-(Euclidean-Method)"]], "A.2. Using Tangent space projection + Logistic Regression": [[43, "A.2.-Using-Tangent-space-projection-+-Logistic-Regression"]], "A.3. Using Tangent space projection + MLP": [[43, "A.3.-Using-Tangent-space-projection-+-MLP"]], "A.4. Using Euclidean MDM": [[43, "A.4.-Using-Euclidean-MDM"]], "A.5. Using Riemannian MDM": [[43, "A.5.-Using-Riemannian-MDM"]], "Summary plots": [[43, "Summary-plots"]], "Hyperbolic Embedding of Graphs and Clustering": [[44, "Hyperbolic-Embedding-of-Graphs-and-Clustering"]], "Parameters and Initialization": [[44, "Parameters-and-Initialization"]], "Loss function.": [[44, "Loss-function."]], "Riemannian optimization.": [[44, "Riemannian-optimization."]], "Capturing the graph structure": [[44, "Capturing-the-graph-structure"]], "Numerically optimizing the loss function": [[44, "Numerically-optimizing-the-loss-function"]], "Plotting results": [[44, "Plotting-results"]], "Classifying hands poses with Kendall shape spaces": [[45, "Classifying-hands-poses-with-Kendall-shape-spaces"]], "Hands shapes": [[45, "Hands-shapes"]], "Computing with shapes of landmarks in Kendall shape spaces": [[46, "Computing-with-shapes-of-landmarks-in-Kendall-shape-spaces"]], "Computing with triangular shapes in Kendall framework": [[47, "Computing-with-triangular-shapes-in-Kendall-framework"]], "The space \\Sigma_2^3 of 2D triangles": [[47, "The-space-\\Sigma_2^3-of-2D-triangles"]], "Uniform distribution": [[47, "Uniform-distribution"], [47, "id1"]], "Geodesic": [[47, "Geodesic"]], "Parallel transport": [[47, "Parallel-transport"], [47, "id2"]], "The space \\Sigma_3^3 of 3D triangles": [[47, "The-space-\\Sigma_3^3-of-3D-triangles"]], "Usecase : optical nerve heads": [[47, "Usecase-:-optical-nerve-heads"]], "Stratified spaces": [[48, "Stratified-spaces"]], "Spider": [[48, "Spider"]], "Graph Space": [[48, "Graph-Space"], [51, "Graph-Space"]], "Graph": [[48, "Graph"]], "Set of Graphs: GraphSpace": [[48, "Set-of-Graphs:-GraphSpace"]], "Labelled or Unlabelled?": [[48, "Labelled-or-Unlabelled?"]], "Graph Space Metric": [[48, "Graph-Space-Metric"]], "Graph to Graph Alignment": [[48, "Graph-to-Graph-Alignment"]], "Graph to Geodesic Alignment": [[48, "Graph-to-Geodesic-Alignment"]], "Optimization of Sao Paulo traffic": [[49, "Optimization-of-Sao-Paulo-traffic"]], "2. Dataset description": [[49, "2.-Dataset-description"]], "Pre-processing the dataset": [[49, "Pre-processing-the-dataset"]], "3. Mathematical modeling": [[49, "3.-Mathematical-modeling"]], "3.1. Road representation: introduction to the Gamma manifold.": [[49, "3.1.-Road-representation:-introduction-to-the-Gamma-manifold."]], "3.1.1. Hypotheses": [[49, "3.1.1.-Hypotheses"]], "3.1.2. Model": [[49, "3.1.2.-Model"]], "3.2. Mathematical representation of renovation efforts": [[49, "3.2.-Mathematical-representation-of-renovation-efforts"]], "3.2.1. Hypotheses": [[49, "3.2.1.-Hypotheses"]], "3.2.2. Model": [[49, "3.2.2.-Model"]], "3.3. Optimization problem": [[49, "3.3.-Optimization-problem"]], "4. Dataset processing": [[49, "4.-Dataset-processing"]], "5. Solving the problem at hand": [[49, "5.-Solving-the-problem-at-hand"]], "Align all and Compute for Graphs": [[50, "Align-all-and-Compute-for-Graphs"]], "A primer in space, metric and aligners": [[50, "A-primer-in-space,-metric-and-aligners"]], "Frechet Mean": [[50, "Frechet-Mean"]], "Principal Components": [[50, "Principal-Components"]], "Regression": [[50, "Regression"]], "Mammals Grooming": [[51, "Mammals-Grooming"]], "Motivation and Dataset": [[51, "Motivation-and-Dataset"], [51, "id1"]], "GGRegression Fifa 2018 Player Passing Networks": [[51, "GGRegression-Fifa-2018-Player-Passing-Networks"]], "Tutorial: Sub-Riemannian geometry on the Heisenberg group": [[52, "Tutorial:-Sub-Riemannian-geometry-on-the-Heisenberg-group"]], "The Heisenberg group": [[52, "The-Heisenberg-group"]], "Defining the Heisenberg sub-Riemannian structure via a frame-field": [[52, "Defining-the-Heisenberg-sub-Riemannian-structure-via-a-frame-field"]], "Sub-Riemannian structures as limits of Riemannian structures": [[52, "Sub-Riemannian-structures-as-limits-of-Riemannian-structures"]], "Roadmap": [[53, "roadmap"]], "Purpose of this document": [[53, "purpose-of-this-document"]], "Statement of purpose: Geomstats in 2022": [[53, "statement-of-purpose-geomstats-in-2022"]], "Detailed Goals": [[53, "detailed-goals"]], "Tutorials": [[54, "tutorials"]], "Practical methods": [[54, "practical-methods"]], "Real world applications": [[54, "real-world-applications"]]}, "indexentries": {"autodiffnotimplementederror": [[0, "geomstats.exceptions.AutodiffNotImplementedError"]], "shapeerror": [[0, "geomstats.errors.ShapeError"]], "adapt_result() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.adapt_result"]], "adapt_types() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.adapt_types"]], "broadcast_to_multibatch() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.broadcast_to_multibatch"]], "check_belongs() (in module geomstats.errors)": [[0, "geomstats.errors.check_belongs"]], "check_integer() (in module geomstats.errors)": [[0, "geomstats.errors.check_integer"]], "check_is_batch() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.check_is_batch"]], "check_parameter_accepted_values() (in module geomstats.errors)": [[0, "geomstats.errors.check_parameter_accepted_values"]], "check_point_shape() (in module geomstats.errors)": [[0, "geomstats.errors.check_point_shape"]], "check_positive() (in module geomstats.errors)": [[0, "geomstats.errors.check_positive"]], "euler_step() (in module geomstats.integrator)": [[0, "geomstats.integrator.euler_step"]], "flip_determinant() (in module geomstats.algebra_utils)": [[0, "geomstats.algebra_utils.flip_determinant"]], "from_vector_to_diagonal_matrix() (in module geomstats.algebra_utils)": [[0, "geomstats.algebra_utils.from_vector_to_diagonal_matrix"]], "geomstats": [[0, "module-geomstats"]], "geomstats.algebra_utils": [[0, "module-geomstats.algebra_utils"]], "geomstats.errors": [[0, "module-geomstats.errors"]], "geomstats.exceptions": [[0, "module-geomstats.exceptions"]], "geomstats.integrator": [[0, "module-geomstats.integrator"]], "geomstats.vectorization": [[0, "module-geomstats.vectorization"]], "get_batch_shape() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.get_batch_shape"]], "get_initial_shapes() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.get_initial_shapes"]], "get_n_points() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.get_n_points"]], "get_types() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.get_types"]], "integrate() (in module geomstats.integrator)": [[0, "geomstats.integrator.integrate"]], "is_scalar() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.is_scalar"]], "leapfrog_step() (in module geomstats.integrator)": [[0, "geomstats.integrator.leapfrog_step"]], "module": [[0, "module-geomstats"], [0, "module-geomstats.algebra_utils"], [0, "module-geomstats.errors"], [0, "module-geomstats.exceptions"], [0, "module-geomstats.integrator"], [0, "module-geomstats.vectorization"], [1, "module-geomstats.datasets"], [1, "module-geomstats.datasets.prepare_emg_data"], [1, "module-geomstats.datasets.prepare_graph_data"], [1, "module-geomstats.datasets.utils"], [2, "module-geomstats.distributions"], [2, "module-geomstats.distributions.lognormal"], [3, "module-geomstats.geometry"], [3, "module-geomstats.geometry.base"], [3, "module-geomstats.geometry.complex_manifold"], [3, "module-geomstats.geometry.complex_matrices"], [3, "module-geomstats.geometry.complex_poincare_disk"], [3, "module-geomstats.geometry.complex_riemannian_metric"], [3, "module-geomstats.geometry.connection"], [3, "module-geomstats.geometry.diffeo"], [3, "module-geomstats.geometry.discrete_curves"], [3, "module-geomstats.geometry.discrete_surfaces"], [3, "module-geomstats.geometry.euclidean"], [3, "module-geomstats.geometry.fiber_bundle"], [3, "module-geomstats.geometry.full_rank_correlation_matrices"], [3, "module-geomstats.geometry.full_rank_matrices"], [3, "module-geomstats.geometry.functions"], [3, "module-geomstats.geometry.general_linear"], [3, "module-geomstats.geometry.grassmannian"], [3, "module-geomstats.geometry.heisenberg"], [3, "module-geomstats.geometry.hermitian"], [3, "module-geomstats.geometry.hermitian_matrices"], [3, "module-geomstats.geometry.hpd_matrices"], [3, "module-geomstats.geometry.hyperbolic"], [3, "module-geomstats.geometry.hyperboloid"], [3, "module-geomstats.geometry.hypersphere"], [3, "module-geomstats.geometry.invariant_metric"], [3, "module-geomstats.geometry.landmarks"], [3, "module-geomstats.geometry.lie_algebra"], [3, "module-geomstats.geometry.lie_group"], [3, "module-geomstats.geometry.lower_triangular_matrices"], [3, "module-geomstats.geometry.manifold"], [3, "module-geomstats.geometry.matrices"], [3, "module-geomstats.geometry.minkowski"], [3, "module-geomstats.geometry.nfold_manifold"], [3, "module-geomstats.geometry.open_hemisphere"], [3, "module-geomstats.geometry.poincare_ball"], [3, "module-geomstats.geometry.poincare_half_space"], [3, "module-geomstats.geometry.poincare_polydisk"], [3, "module-geomstats.geometry.positive_lower_triangular_matrices"], [3, "module-geomstats.geometry.positive_reals"], [3, "module-geomstats.geometry.pre_shape"], [3, "module-geomstats.geometry.product_hpd_and_siegel_disks"], [3, "module-geomstats.geometry.product_manifold"], [3, "module-geomstats.geometry.product_positive_reals_and_poincare_disks"], [3, "module-geomstats.geometry.pullback_metric"], [3, "module-geomstats.geometry.quotient_metric"], [3, "module-geomstats.geometry.rank_k_psd_matrices"], [3, "module-geomstats.geometry.riemannian_metric"], [3, "module-geomstats.geometry.sasaki_metric"], [3, "module-geomstats.geometry.scalar_product_metric"], [3, "module-geomstats.geometry.siegel"], [3, "module-geomstats.geometry.skew_symmetric_matrices"], [3, "module-geomstats.geometry.spd_matrices"], [3, "module-geomstats.geometry.special_euclidean"], [3, "module-geomstats.geometry.special_orthogonal"], [3, "module-geomstats.geometry.stiefel"], [3, "module-geomstats.geometry.sub_riemannian_metric"], [3, "module-geomstats.geometry.symmetric_matrices"], [4, "module-geomstats.geometry.stratified"], [4, "module-geomstats.geometry.stratified.graph_space"], [4, "module-geomstats.geometry.stratified.point_set"], [4, "module-geomstats.geometry.stratified.spider"], [4, "module-geomstats.geometry.stratified.wald_space"], [5, "module-geomstats.information_geometry"], [5, "module-geomstats.information_geometry.base"], [5, "module-geomstats.information_geometry.beta"], [5, "module-geomstats.information_geometry.binomial"], [5, "module-geomstats.information_geometry.categorical"], [5, "module-geomstats.information_geometry.dirichlet"], [5, "module-geomstats.information_geometry.exponential"], [5, "module-geomstats.information_geometry.fisher_rao_metric"], [5, "module-geomstats.information_geometry.gamma"], [5, "module-geomstats.information_geometry.geometric"], [5, "module-geomstats.information_geometry.multinomial"], [5, "module-geomstats.information_geometry.normal"], [5, "module-geomstats.information_geometry.poisson"], [6, "module-geomstats.learning"], [6, "module-geomstats.learning.aac"], [6, "module-geomstats.learning.agglomerative_hierarchical_clustering"], [6, "module-geomstats.learning.expectation_maximization"], [6, "module-geomstats.learning.exponential_barycenter"], [6, "module-geomstats.learning.frechet_mean"], [6, "module-geomstats.learning.geodesic_regression"], [6, "module-geomstats.learning.geometric_median"], [6, "module-geomstats.learning.incremental_frechet_mean"], [6, "module-geomstats.learning.kalman_filter"], [6, "module-geomstats.learning.kernel_density_estimation_classifier"], [6, "module-geomstats.learning.kmeans"], [6, "module-geomstats.learning.kmedoids"], [6, "module-geomstats.learning.knn"], [6, "module-geomstats.learning.mdm"], [6, "module-geomstats.learning.online_kmeans"], [6, "module-geomstats.learning.pca"], [6, "module-geomstats.learning.preprocessing"], [6, "module-geomstats.learning.radial_kernel_functions"], [6, "module-geomstats.learning.riemannian_mean_shift"], [6, "module-geomstats.learning.wrapped_gaussian_process"], [7, "module-geomstats.numerics"], [7, "module-geomstats.numerics.bvp"], [7, "module-geomstats.numerics.geodesic"], [7, "module-geomstats.numerics.ivp"], [7, "module-geomstats.numerics.optimizers"], [8, "module-geomstats.test"], [8, "module-geomstats.test.data"], [8, "module-geomstats.test.random"], [8, "module-geomstats.test.utils"], [8, "module-geomstats.test.vectorization"], [9, "module-geomstats.test_cases"], [11, "module-geomstats.test_cases.datasets"], [12, "module-geomstats.test_cases.distributions"], [13, "module-geomstats.test_cases.geometry"], [14, "module-geomstats.test_cases.geometry.stratified"], [15, "module-geomstats.test_cases.information_geometry"], [16, "module-geomstats.test_cases.learning"], [16, "module-geomstats.test_cases.learning.exponential_barycenter"], [17, "module-geomstats.test_cases.numerics"], [18, "module-geomstats.visualization"], [18, "module-geomstats.visualization.hyperbolic"], [18, "module-geomstats.visualization.hypersphere"], [18, "module-geomstats.visualization.poincare_polydisk"], [18, "module-geomstats.visualization.pre_shape"], [18, "module-geomstats.visualization.spd_matrices"], [18, "module-geomstats.visualization.special_euclidean"], [18, "module-geomstats.visualization.special_orthogonal"]], "repeat_out() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.repeat_out"]], "repeat_point() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.repeat_point"]], "rk2_step() (in module geomstats.integrator)": [[0, "geomstats.integrator.rk2_step"]], "rk4_step() (in module geomstats.integrator)": [[0, "geomstats.integrator.rk4_step"]], "rotate_points() (in module geomstats.algebra_utils)": [[0, "geomstats.algebra_utils.rotate_points"]], "squeeze_output_dim_0() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.squeeze_output_dim_0"]], "squeeze_output_dim_1() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.squeeze_output_dim_1"]], "symplectic_euler_step() (in module geomstats.integrator)": [[0, "geomstats.integrator.symplectic_euler_step"]], "taylor_exp_even_func() (in module geomstats.algebra_utils)": [[0, "geomstats.algebra_utils.taylor_exp_even_func"]], "vectorize_args() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.vectorize_args"]], "vectorize_kwargs() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.vectorize_kwargs"]], "graph (class in geomstats.datasets.prepare_graph_data)": [[1, "geomstats.datasets.prepare_graph_data.Graph"]], "hyperbolicembedding (class in geomstats.datasets.prepare_graph_data)": [[1, "geomstats.datasets.prepare_graph_data.HyperbolicEmbedding"]], "timeseriescovariance (class in geomstats.datasets.prepare_emg_data)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance"]], "batches (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.batches"]], "covec (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.covec"]], "covs (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.covs"]], "data_dict (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.data_dict"]], "diags (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.diags"]], "edges (geomstats.datasets.prepare_graph_data.graph attribute)": [[1, "geomstats.datasets.prepare_graph_data.Graph.edges"]], "embed() (geomstats.datasets.prepare_graph_data.hyperbolicembedding method)": [[1, "geomstats.datasets.prepare_graph_data.HyperbolicEmbedding.embed"]], "geomstats.datasets": [[1, "module-geomstats.datasets"]], "geomstats.datasets.prepare_emg_data": [[1, "module-geomstats.datasets.prepare_emg_data"]], "geomstats.datasets.prepare_graph_data": [[1, "module-geomstats.datasets.prepare_graph_data"]], "geomstats.datasets.utils": [[1, "module-geomstats.datasets.utils"]], "grad_log_sigmoid() (geomstats.datasets.prepare_graph_data.hyperbolicembedding static method)": [[1, "geomstats.datasets.prepare_graph_data.HyperbolicEmbedding.grad_log_sigmoid"]], "grad_squared_distance() (geomstats.datasets.prepare_graph_data.hyperbolicembedding method)": [[1, "geomstats.datasets.prepare_graph_data.HyperbolicEmbedding.grad_squared_distance"]], "label_map (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.label_map"]], "labels (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.labels"]], "labels (geomstats.datasets.prepare_graph_data.graph attribute)": [[1, "geomstats.datasets.prepare_graph_data.Graph.labels"]], "load_cells() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_cells"]], "load_cities() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_cities"]], "load_connectomes() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_connectomes"]], "load_cube() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_cube"]], "load_emg() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_emg"]], "load_football() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_football"]], "load_hands() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_hands"]], "load_karate_graph() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_karate_graph"]], "load_leaves() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_leaves"]], "load_mammals() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_mammals"]], "load_optical_nerves() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_optical_nerves"]], "load_poses() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_poses"]], "load_random_graph() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_random_graph"]], "load_sao_paulo() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_sao_paulo"]], "log_sigmoid() (geomstats.datasets.prepare_graph_data.hyperbolicembedding static method)": [[1, "geomstats.datasets.prepare_graph_data.HyperbolicEmbedding.log_sigmoid"]], "loss() (geomstats.datasets.prepare_graph_data.hyperbolicembedding method)": [[1, "geomstats.datasets.prepare_graph_data.HyperbolicEmbedding.loss"]], "margin (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.margin"]], "n_nodes (geomstats.datasets.prepare_graph_data.graph attribute)": [[1, "geomstats.datasets.prepare_graph_data.Graph.n_nodes"]], "n_steps (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.n_steps"]], "n_timeseries (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.n_timeseries"]], "random_walk() (geomstats.datasets.prepare_graph_data.graph method)": [[1, 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"geomstats.distributions.lognormal.LogNormalSPD.sample"]], "samples_sym() (geomstats.distributions.lognormal.lognormalspd method)": [[2, "geomstats.distributions.lognormal.LogNormalSPD.samples_sym"]], "autodiffdiffeo (class in geomstats.geometry.diffeo)": [[3, "geomstats.geometry.diffeo.AutodiffDiffeo"]], "biinvariantmetric (class in geomstats.geometry.invariant_metric)": [[3, "geomstats.geometry.invariant_metric.BiInvariantMetric"]], "bureswassersteinbundle (class in geomstats.geometry.rank_k_psd_matrices)": [[3, "geomstats.geometry.rank_k_psd_matrices.BuresWassersteinBundle"]], "choleskymap (class in geomstats.geometry.spd_matrices)": [[3, "geomstats.geometry.spd_matrices.CholeskyMap"]], "choleskymetric (class in geomstats.geometry.positive_lower_triangular_matrices)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.CholeskyMetric"]], "complexmanifold (class in geomstats.geometry.complex_manifold)": [[3, "geomstats.geometry.complex_manifold.ComplexManifold"]], 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(geomstats.geometry.pre_shape.preshapemetric method)": [[3, "geomstats.geometry.pre_shape.PreShapeMetric.exp"]], "exp() (geomstats.geometry.product_manifold.productriemannianmetric method)": [[3, "geomstats.geometry.product_manifold.ProductRiemannianMetric.exp"]], "exp() (geomstats.geometry.pullback_metric.pullbackdiffeometric method)": [[3, "geomstats.geometry.pullback_metric.PullbackDiffeoMetric.exp"]], "exp() (geomstats.geometry.quotient_metric.quotientmetric method)": [[3, "geomstats.geometry.quotient_metric.QuotientMetric.exp"]], "exp() (geomstats.geometry.sasaki_metric.sasakimetric method)": [[3, "geomstats.geometry.sasaki_metric.SasakiMetric.exp"]], "exp() (geomstats.geometry.siegel.siegelmetric method)": [[3, "geomstats.geometry.siegel.SiegelMetric.exp"]], "exp() (geomstats.geometry.spd_matrices.spdaffinemetric method)": [[3, "geomstats.geometry.spd_matrices.SPDAffineMetric.exp"]], "exp() (geomstats.geometry.spd_matrices.spdbureswassersteinmetric method)": [[3, "geomstats.geometry.spd_matrices.SPDBuresWassersteinMetric.exp"]], "exp() (geomstats.geometry.special_euclidean.specialeuclideanmatricescanonicalleftmetric method)": [[3, "geomstats.geometry.special_euclidean.SpecialEuclideanMatricesCanonicalLeftMetric.exp"]], "exp() (geomstats.geometry.stiefel.stiefelcanonicalmetric method)": [[3, "geomstats.geometry.stiefel.StiefelCanonicalMetric.exp"]], "exp() (geomstats.geometry.sub_riemannian_metric.subriemannianmetric method)": [[3, "geomstats.geometry.sub_riemannian_metric.SubRiemannianMetric.exp"]], "exp_at_zero() (geomstats.geometry.siegel.siegelmetric static method)": [[3, "geomstats.geometry.siegel.SiegelMetric.exp_at_zero"]], "exp_domain() (geomstats.geometry.hpd_matrices.hpdeuclideanmetric static method)": [[3, "geomstats.geometry.hpd_matrices.HPDEuclideanMetric.exp_domain"]], "exp_domain() (geomstats.geometry.spd_matrices.spdeuclideanmetric static method)": [[3, "geomstats.geometry.spd_matrices.SPDEuclideanMetric.exp_domain"]], "exp_from_identity() (geomstats.geometry.heisenberg.heisenbergvectors method)": [[3, "geomstats.geometry.heisenberg.HeisenbergVectors.exp_from_identity"]], "exp_from_identity() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.exp_from_identity"]], "exp_not_from_identity() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.exp_not_from_identity"]], "expmh() (in module geomstats.geometry.hermitian_matrices)": [[3, "geomstats.geometry.hermitian_matrices.expmh"]], "extrinsic_to_intrinsic_coords() (geomstats.geometry.base.levelset method)": [[3, "geomstats.geometry.base.LevelSet.extrinsic_to_intrinsic_coords"]], "extrinsic_to_intrinsic_coords() (geomstats.geometry.hyperboloid.hyperboloid method)": [[3, "geomstats.geometry.hyperboloid.Hyperboloid.extrinsic_to_intrinsic_coords"]], "face_areas() (geomstats.geometry.discrete_surfaces.discretesurfaces method)": [[3, "geomstats.geometry.discrete_surfaces.DiscreteSurfaces.face_areas"]], "flatten() (geomstats.geometry.matrices.matrices static method)": [[3, "geomstats.geometry.matrices.Matrices.flatten"]], "forward_difference() (in module geomstats.geometry.discrete_curves)": [[3, "geomstats.geometry.discrete_curves.forward_difference"]], "frobenius_product() (geomstats.geometry.complex_matrices.complexmatrices static method)": [[3, "geomstats.geometry.complex_matrices.ComplexMatrices.frobenius_product"]], "frobenius_product() (geomstats.geometry.matrices.matrices static method)": [[3, "geomstats.geometry.matrices.Matrices.frobenius_product"]], "from_covariance() (geomstats.geometry.full_rank_correlation_matrices.fullrankcorrelationmatrices class method)": [[3, "geomstats.geometry.full_rank_correlation_matrices.FullRankCorrelationMatrices.from_covariance"]], "geodesic() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.geodesic"]], "geodesic() (geomstats.geometry.product_manifold.productriemannianmetric method)": [[3, "geomstats.geometry.product_manifold.ProductRiemannianMetric.geodesic"]], "geodesic() (geomstats.geometry.pullback_metric.pullbackdiffeometric method)": [[3, "geomstats.geometry.pullback_metric.PullbackDiffeoMetric.geodesic"]], "geodesic() (geomstats.geometry.quotient_metric.quotientmetric method)": [[3, "geomstats.geometry.quotient_metric.QuotientMetric.geodesic"]], "geodesic() (geomstats.geometry.special_euclidean.specialeuclideanmatricescanonicalleftmetric method)": [[3, "geomstats.geometry.special_euclidean.SpecialEuclideanMatricesCanonicalLeftMetric.geodesic"]], "geodesic() (geomstats.geometry.sub_riemannian_metric.subriemannianmetric method)": [[3, "geomstats.geometry.sub_riemannian_metric.SubRiemannianMetric.geodesic"]], "geodesic_discrete() (geomstats.geometry.sasaki_metric.sasakimetric method)": [[3, "geomstats.geometry.sasaki_metric.SasakiMetric.geodesic_discrete"]], "geodesic_equation() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.geodesic_equation"]], "geomstats.geometry": [[3, "module-geomstats.geometry"]], "geomstats.geometry.base": [[3, "module-geomstats.geometry.base"]], "geomstats.geometry.complex_manifold": [[3, "module-geomstats.geometry.complex_manifold"]], "geomstats.geometry.complex_matrices": [[3, "module-geomstats.geometry.complex_matrices"]], "geomstats.geometry.complex_poincare_disk": [[3, "module-geomstats.geometry.complex_poincare_disk"]], "geomstats.geometry.complex_riemannian_metric": [[3, "module-geomstats.geometry.complex_riemannian_metric"]], "geomstats.geometry.connection": [[3, "module-geomstats.geometry.connection"]], "geomstats.geometry.diffeo": [[3, "module-geomstats.geometry.diffeo"]], "geomstats.geometry.discrete_curves": [[3, "module-geomstats.geometry.discrete_curves"]], "geomstats.geometry.discrete_surfaces": [[3, "module-geomstats.geometry.discrete_surfaces"]], "geomstats.geometry.euclidean": [[3, "module-geomstats.geometry.euclidean"]], "geomstats.geometry.fiber_bundle": [[3, "module-geomstats.geometry.fiber_bundle"]], "geomstats.geometry.full_rank_correlation_matrices": [[3, "module-geomstats.geometry.full_rank_correlation_matrices"]], "geomstats.geometry.full_rank_matrices": [[3, "module-geomstats.geometry.full_rank_matrices"]], "geomstats.geometry.functions": [[3, "module-geomstats.geometry.functions"]], "geomstats.geometry.general_linear": [[3, "module-geomstats.geometry.general_linear"]], "geomstats.geometry.grassmannian": [[3, "module-geomstats.geometry.grassmannian"]], "geomstats.geometry.heisenberg": [[3, "module-geomstats.geometry.heisenberg"]], "geomstats.geometry.hermitian": [[3, "module-geomstats.geometry.hermitian"]], "geomstats.geometry.hermitian_matrices": [[3, "module-geomstats.geometry.hermitian_matrices"]], "geomstats.geometry.hpd_matrices": [[3, "module-geomstats.geometry.hpd_matrices"]], "geomstats.geometry.hyperbolic": [[3, "module-geomstats.geometry.hyperbolic"]], "geomstats.geometry.hyperboloid": [[3, "module-geomstats.geometry.hyperboloid"]], "geomstats.geometry.hypersphere": [[3, "module-geomstats.geometry.hypersphere"]], "geomstats.geometry.invariant_metric": [[3, "module-geomstats.geometry.invariant_metric"]], "geomstats.geometry.landmarks": [[3, "module-geomstats.geometry.landmarks"]], "geomstats.geometry.lie_algebra": [[3, "module-geomstats.geometry.lie_algebra"]], "geomstats.geometry.lie_group": [[3, "module-geomstats.geometry.lie_group"]], "geomstats.geometry.lower_triangular_matrices": [[3, "module-geomstats.geometry.lower_triangular_matrices"]], "geomstats.geometry.manifold": [[3, "module-geomstats.geometry.manifold"]], "geomstats.geometry.matrices": [[3, "module-geomstats.geometry.matrices"]], "geomstats.geometry.minkowski": [[3, "module-geomstats.geometry.minkowski"]], "geomstats.geometry.nfold_manifold": [[3, "module-geomstats.geometry.nfold_manifold"]], "geomstats.geometry.open_hemisphere": [[3, "module-geomstats.geometry.open_hemisphere"]], "geomstats.geometry.poincare_ball": [[3, "module-geomstats.geometry.poincare_ball"]], "geomstats.geometry.poincare_half_space": [[3, "module-geomstats.geometry.poincare_half_space"]], "geomstats.geometry.poincare_polydisk": [[3, "module-geomstats.geometry.poincare_polydisk"]], "geomstats.geometry.positive_lower_triangular_matrices": [[3, "module-geomstats.geometry.positive_lower_triangular_matrices"]], "geomstats.geometry.positive_reals": [[3, "module-geomstats.geometry.positive_reals"]], "geomstats.geometry.pre_shape": [[3, "module-geomstats.geometry.pre_shape"]], "geomstats.geometry.product_hpd_and_siegel_disks": [[3, "module-geomstats.geometry.product_hpd_and_siegel_disks"]], "geomstats.geometry.product_manifold": [[3, "module-geomstats.geometry.product_manifold"]], "geomstats.geometry.product_positive_reals_and_poincare_disks": [[3, "module-geomstats.geometry.product_positive_reals_and_poincare_disks"]], "geomstats.geometry.pullback_metric": [[3, "module-geomstats.geometry.pullback_metric"]], "geomstats.geometry.quotient_metric": [[3, "module-geomstats.geometry.quotient_metric"]], "geomstats.geometry.rank_k_psd_matrices": [[3, "module-geomstats.geometry.rank_k_psd_matrices"]], "geomstats.geometry.riemannian_metric": [[3, "module-geomstats.geometry.riemannian_metric"]], "geomstats.geometry.sasaki_metric": [[3, "module-geomstats.geometry.sasaki_metric"]], "geomstats.geometry.scalar_product_metric": [[3, "module-geomstats.geometry.scalar_product_metric"]], "geomstats.geometry.siegel": [[3, "module-geomstats.geometry.siegel"]], "geomstats.geometry.skew_symmetric_matrices": [[3, "module-geomstats.geometry.skew_symmetric_matrices"]], "geomstats.geometry.spd_matrices": [[3, "module-geomstats.geometry.spd_matrices"]], "geomstats.geometry.special_euclidean": [[3, "module-geomstats.geometry.special_euclidean"]], "geomstats.geometry.special_orthogonal": [[3, "module-geomstats.geometry.special_orthogonal"]], "geomstats.geometry.stiefel": [[3, "module-geomstats.geometry.stiefel"]], "geomstats.geometry.sub_riemannian_metric": [[3, "module-geomstats.geometry.sub_riemannian_metric"]], "geomstats.geometry.symmetric_matrices": [[3, "module-geomstats.geometry.symmetric_matrices"]], "hamiltonian() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.hamiltonian"]], "hamiltonian() (geomstats.geometry.sub_riemannian_metric.subriemannianmetric method)": [[3, "geomstats.geometry.sub_riemannian_metric.SubRiemannianMetric.hamiltonian"]], "hessian_immersion() (geomstats.geometry.base.immersedset method)": [[3, "geomstats.geometry.base.ImmersedSet.hessian_immersion"]], "homogeneous_representation() (in module geomstats.geometry.special_euclidean)": [[3, "geomstats.geometry.special_euclidean.homogeneous_representation"]], "horizontal_lift() (geomstats.geometry.fiber_bundle.fiberbundle method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.horizontal_lift"]], "horizontal_lift() (geomstats.geometry.full_rank_correlation_matrices.correlationmatricesbundle method)": [[3, "geomstats.geometry.full_rank_correlation_matrices.CorrelationMatricesBundle.horizontal_lift"]], "horizontal_lift() (geomstats.geometry.rank_k_psd_matrices.bureswassersteinbundle method)": [[3, "geomstats.geometry.rank_k_psd_matrices.BuresWassersteinBundle.horizontal_lift"]], "horizontal_projection() (geomstats.geometry.discrete_curves.srvreparametrizationbundle method)": [[3, "geomstats.geometry.discrete_curves.SRVReparametrizationBundle.horizontal_projection"]], "horizontal_projection() (geomstats.geometry.discrete_curves.srvrotationbundle method)": [[3, "geomstats.geometry.discrete_curves.SRVRotationBundle.horizontal_projection"]], "horizontal_projection() (geomstats.geometry.discrete_curves.srvrotationreparametrizationbundle method)": [[3, "geomstats.geometry.discrete_curves.SRVRotationReparametrizationBundle.horizontal_projection"]], "horizontal_projection() (geomstats.geometry.fiber_bundle.fiberbundle method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.horizontal_projection"]], "identity (geomstats.geometry.euclidean.euclidean property)": [[3, "geomstats.geometry.euclidean.Euclidean.identity"]], "identity (geomstats.geometry.heisenberg.heisenbergvectors property)": [[3, "geomstats.geometry.heisenberg.HeisenbergVectors.identity"]], "identity (geomstats.geometry.hermitian.hermitian property)": [[3, "geomstats.geometry.hermitian.Hermitian.identity"]], "identity (geomstats.geometry.lie_group.liegroup property)": [[3, "geomstats.geometry.lie_group.LieGroup.identity"]], "identity (geomstats.geometry.lie_group.matrixliegroup property)": [[3, "geomstats.geometry.lie_group.MatrixLieGroup.identity"]], "immersion() (geomstats.geometry.base.immersedset method)": [[3, "geomstats.geometry.base.ImmersedSet.immersion"]], "injectivity_radius() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.grassmannian.grassmanniancanonicalmetric method)": [[3, "geomstats.geometry.grassmannian.GrassmannianCanonicalMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.hpd_matrices.hpdaffinemetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDAffineMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.hpd_matrices.hpdbureswassersteinmetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDBuresWassersteinMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.hpd_matrices.hpdeuclideanmetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDEuclideanMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.hyperboloid.hyperboloidmetric method)": [[3, "geomstats.geometry.hyperboloid.HyperboloidMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, "geomstats.geometry.hypersphere.HypersphereMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.invariant_metric.biinvariantmetric method)": [[3, "geomstats.geometry.invariant_metric.BiInvariantMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.minkowski.minkowskimetric method)": [[3, "geomstats.geometry.minkowski.MinkowskiMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.poincare_half_space.poincarehalfspacemetric method)": [[3, "geomstats.geometry.poincare_half_space.PoincareHalfSpaceMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.pre_shape.preshapemetric method)": [[3, "geomstats.geometry.pre_shape.PreShapeMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.spd_matrices.spdaffinemetric method)": [[3, "geomstats.geometry.spd_matrices.SPDAffineMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.spd_matrices.spdbureswassersteinmetric method)": [[3, "geomstats.geometry.spd_matrices.SPDBuresWassersteinMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.spd_matrices.spdeuclideanmetric method)": [[3, "geomstats.geometry.spd_matrices.SPDEuclideanMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.special_euclidean.specialeuclideanmatricescanonicalleftmetric method)": [[3, "geomstats.geometry.special_euclidean.SpecialEuclideanMatricesCanonicalLeftMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.stiefel.stiefelcanonicalmetric method)": [[3, "geomstats.geometry.stiefel.StiefelCanonicalMetric.injectivity_radius"]], "inner_coproduct() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.inner_coproduct"]], "inner_coproduct() (geomstats.geometry.sub_riemannian_metric.subriemannianmetric method)": [[3, "geomstats.geometry.sub_riemannian_metric.SubRiemannianMetric.inner_coproduct"]], "inner_product() (geomstats.geometry.complex_matrices.complexmatricesmetric static method)": [[3, "geomstats.geometry.complex_matrices.ComplexMatricesMetric.inner_product"]], "inner_product() (geomstats.geometry.complex_riemannian_metric.complexriemannianmetric method)": [[3, "geomstats.geometry.complex_riemannian_metric.ComplexRiemannianMetric.inner_product"]], "inner_product() (geomstats.geometry.discrete_curves.l2curvesmetric method)": [[3, "geomstats.geometry.discrete_curves.L2CurvesMetric.inner_product"]], "inner_product() (geomstats.geometry.discrete_surfaces.elasticmetric method)": [[3, "geomstats.geometry.discrete_surfaces.ElasticMetric.inner_product"]], "inner_product() (geomstats.geometry.euclidean.euclideanmetric method)": [[3, "geomstats.geometry.euclidean.EuclideanMetric.inner_product"]], "inner_product() (geomstats.geometry.functions.hilbertspheremetric method)": [[3, "geomstats.geometry.functions.HilbertSphereMetric.inner_product"]], "inner_product() (geomstats.geometry.hermitian.hermitianmetric static method)": [[3, "geomstats.geometry.hermitian.HermitianMetric.inner_product"]], "inner_product() (geomstats.geometry.hpd_matrices.hpdaffinemetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDAffineMetric.inner_product"]], "inner_product() (geomstats.geometry.hpd_matrices.hpdbureswassersteinmetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDBuresWassersteinMetric.inner_product"]], "inner_product() (geomstats.geometry.hyperboloid.hyperboloidmetric method)": [[3, "geomstats.geometry.hyperboloid.HyperboloidMetric.inner_product"]], "inner_product() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, "geomstats.geometry.hypersphere.HypersphereMetric.inner_product"]], "inner_product() (geomstats.geometry.invariant_metric.biinvariantmetric method)": [[3, "geomstats.geometry.invariant_metric.BiInvariantMetric.inner_product"]], "inner_product() (geomstats.geometry.matrices.matricesmetric method)": [[3, "geomstats.geometry.matrices.MatricesMetric.inner_product"]], "inner_product() (geomstats.geometry.minkowski.minkowskimetric method)": [[3, "geomstats.geometry.minkowski.MinkowskiMetric.inner_product"]], "inner_product() (geomstats.geometry.nfold_manifold.nfoldmetric method)": [[3, "geomstats.geometry.nfold_manifold.NFoldMetric.inner_product"]], "inner_product() (geomstats.geometry.poincare_half_space.poincarehalfspacemetric method)": [[3, "geomstats.geometry.poincare_half_space.PoincareHalfSpaceMetric.inner_product"]], "inner_product() (geomstats.geometry.positive_lower_triangular_matrices.choleskymetric class method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.CholeskyMetric.inner_product"]], "inner_product() (geomstats.geometry.pre_shape.preshapemetric method)": [[3, "geomstats.geometry.pre_shape.PreShapeMetric.inner_product"]], "inner_product() (geomstats.geometry.product_manifold.productriemannianmetric method)": [[3, "geomstats.geometry.product_manifold.ProductRiemannianMetric.inner_product"]], "inner_product() (geomstats.geometry.pullback_metric.pullbackdiffeometric method)": [[3, "geomstats.geometry.pullback_metric.PullbackDiffeoMetric.inner_product"]], "inner_product() (geomstats.geometry.quotient_metric.quotientmetric method)": [[3, "geomstats.geometry.quotient_metric.QuotientMetric.inner_product"]], "inner_product() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.inner_product"]], "inner_product() (geomstats.geometry.sasaki_metric.sasakimetric method)": [[3, "geomstats.geometry.sasaki_metric.SasakiMetric.inner_product"]], "inner_product() (geomstats.geometry.siegel.siegelmetric method)": [[3, "geomstats.geometry.siegel.SiegelMetric.inner_product"]], "inner_product() (geomstats.geometry.spd_matrices.spdaffinemetric method)": [[3, "geomstats.geometry.spd_matrices.SPDAffineMetric.inner_product"]], "inner_product() (geomstats.geometry.spd_matrices.spdbureswassersteinmetric method)": [[3, "geomstats.geometry.spd_matrices.SPDBuresWassersteinMetric.inner_product"]], "inner_product() (geomstats.geometry.special_euclidean.specialeuclideanmatricescanonicalleftmetric method)": [[3, "geomstats.geometry.special_euclidean.SpecialEuclideanMatricesCanonicalLeftMetric.inner_product"]], "inner_product() (geomstats.geometry.stiefel.stiefelcanonicalmetric method)": [[3, "geomstats.geometry.stiefel.StiefelCanonicalMetric.inner_product"]], "inner_product_at_identity() (geomstats.geometry.invariant_metric.biinvariantmetric method)": [[3, "geomstats.geometry.invariant_metric.BiInvariantMetric.inner_product_at_identity"]], "inner_product_at_identity() (geomstats.geometry.positive_lower_triangular_matrices.invariantpositivelowertriangularmatricesmetric method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.InvariantPositiveLowerTriangularMatricesMetric.inner_product_at_identity"]], "inner_product_derivative_matrix() (geomstats.geometry.pullback_metric.pullbackmetric method)": [[3, "geomstats.geometry.pullback_metric.PullbackMetric.inner_product_derivative_matrix"]], "inner_product_derivative_matrix() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.inner_product_derivative_matrix"]], "insert_origin() (geomstats.geometry.discrete_curves.discretecurvesstartingatorigin method)": [[3, "geomstats.geometry.discrete_curves.DiscreteCurvesStartingAtOrigin.insert_origin"]], "insert_zeros() (in module geomstats.geometry.discrete_curves)": [[3, "geomstats.geometry.discrete_curves.insert_zeros"]], "integrability_tensor() (geomstats.geometry.fiber_bundle.fiberbundle method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.integrability_tensor"]], "integrability_tensor() (geomstats.geometry.pre_shape.preshapespacebundle method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpaceBundle.integrability_tensor"]], "integrability_tensor_derivative() (geomstats.geometry.fiber_bundle.fiberbundle method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.integrability_tensor_derivative"]], "integrability_tensor_derivative() (geomstats.geometry.pre_shape.preshapespacebundle method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpaceBundle.integrability_tensor_derivative"]], "integrability_tensor_derivative_parallel() (geomstats.geometry.pre_shape.preshapespacebundle method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpaceBundle.integrability_tensor_derivative_parallel"]], "interpolate() (geomstats.geometry.discrete_curves.discretecurvesstartingatorigin method)": [[3, "geomstats.geometry.discrete_curves.DiscreteCurvesStartingAtOrigin.interpolate"]], "intrinsic_to_extrinsic_coords() (geomstats.geometry.base.levelset method)": [[3, "geomstats.geometry.base.LevelSet.intrinsic_to_extrinsic_coords"]], "intrinsic_to_extrinsic_coords() (geomstats.geometry.hyperboloid.hyperboloid method)": [[3, "geomstats.geometry.hyperboloid.Hyperboloid.intrinsic_to_extrinsic_coords"]], "intrinsic_to_extrinsic_coords() (geomstats.geometry.poincare_polydisk.poincarepolydisk static method)": [[3, "geomstats.geometry.poincare_polydisk.PoincarePolydisk.intrinsic_to_extrinsic_coords"]], "inverse() (geomstats.geometry.heisenberg.heisenbergvectors method)": [[3, "geomstats.geometry.heisenberg.HeisenbergVectors.inverse"]], "inverse() (geomstats.geometry.lie_group.liegroup class method)": [[3, "geomstats.geometry.lie_group.LieGroup.inverse"]], "inverse() (geomstats.geometry.lie_group.matrixliegroup class method)": [[3, "geomstats.geometry.lie_group.MatrixLieGroup.inverse"]], "inverse_diffeomorphism() (geomstats.geometry.diffeo.composeddiffeo method)": [[3, "geomstats.geometry.diffeo.ComposedDiffeo.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.diffeo.diffeo method)": [[3, "geomstats.geometry.diffeo.Diffeo.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.diffeo.reverseddiffeo method)": [[3, "geomstats.geometry.diffeo.ReversedDiffeo.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.discrete_curves.ftransform method)": [[3, "geomstats.geometry.discrete_curves.FTransform.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.discrete_curves.srvtransform method)": [[3, "geomstats.geometry.discrete_curves.SRVTransform.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.open_hemisphere.openhemispheretohyperboloiddiffeo method)": [[3, "geomstats.geometry.open_hemisphere.OpenHemisphereToHyperboloidDiffeo.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.positive_lower_triangular_matrices.unitnormedrowspltdiffeo method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.UnitNormedRowsPLTDiffeo.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.spd_matrices.choleskymap static method)": [[3, "geomstats.geometry.spd_matrices.CholeskyMap.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.spd_matrices.matrixpower method)": [[3, "geomstats.geometry.spd_matrices.MatrixPower.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.spd_matrices.symmatrixlog class method)": [[3, "geomstats.geometry.spd_matrices.SymMatrixLog.inverse_diffeomorphism"]], "inverse_jacobian_diffeomorphism() (geomstats.geometry.diffeo.autodiffdiffeo method)": [[3, "geomstats.geometry.diffeo.AutodiffDiffeo.inverse_jacobian_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.diffeo.autodiffdiffeo method)": [[3, "geomstats.geometry.diffeo.AutodiffDiffeo.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.diffeo.composeddiffeo method)": [[3, "geomstats.geometry.diffeo.ComposedDiffeo.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.diffeo.diffeo method)": [[3, "geomstats.geometry.diffeo.Diffeo.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.diffeo.reverseddiffeo method)": [[3, "geomstats.geometry.diffeo.ReversedDiffeo.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.discrete_curves.srvtransform method)": [[3, "geomstats.geometry.discrete_curves.SRVTransform.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.open_hemisphere.openhemispheretohyperboloiddiffeo method)": [[3, "geomstats.geometry.open_hemisphere.OpenHemisphereToHyperboloidDiffeo.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.positive_lower_triangular_matrices.unitnormedrowspltdiffeo method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.UnitNormedRowsPLTDiffeo.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.spd_matrices.choleskymap class method)": [[3, "geomstats.geometry.spd_matrices.CholeskyMap.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.spd_matrices.matrixpower method)": [[3, "geomstats.geometry.spd_matrices.MatrixPower.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.spd_matrices.symmatrixlog class method)": [[3, "geomstats.geometry.spd_matrices.SymMatrixLog.inverse_tangent_diffeomorphism"]], "is_centered() (geomstats.geometry.pre_shape.preshapespace static method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpace.is_centered"]], "is_diagonal() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_diagonal"]], "is_hermitian() (geomstats.geometry.complex_matrices.complexmatrices class method)": [[3, "geomstats.geometry.complex_matrices.ComplexMatrices.is_hermitian"]], "is_horizontal() (geomstats.geometry.fiber_bundle.fiberbundle method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.is_horizontal"]], "is_horizontal() (geomstats.geometry.pre_shape.preshapespacebundle method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpaceBundle.is_horizontal"]], "is_hpd() (geomstats.geometry.complex_matrices.complexmatrices class method)": [[3, "geomstats.geometry.complex_matrices.ComplexMatrices.is_hpd"]], "is_lower_triangular() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_lower_triangular"]], "is_pd() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_pd"]], "is_skew_hermitian() (geomstats.geometry.complex_matrices.complexmatrices class method)": [[3, "geomstats.geometry.complex_matrices.ComplexMatrices.is_skew_hermitian"]], "is_skew_symmetric() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_skew_symmetric"]], "is_spd() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_spd"]], "is_square() (geomstats.geometry.matrices.matrices static method)": [[3, "geomstats.geometry.matrices.Matrices.is_square"]], "is_strictly_lower_triangular() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_strictly_lower_triangular"]], "is_strictly_upper_triangular() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_strictly_upper_triangular"]], "is_symmetric() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_symmetric"]], "is_tangent() (geomstats.geometry.base.complexvectorspace method)": [[3, "geomstats.geometry.base.ComplexVectorSpace.is_tangent"]], "is_tangent() (geomstats.geometry.base.complexvectorspaceopenset method)": [[3, "geomstats.geometry.base.ComplexVectorSpaceOpenSet.is_tangent"]], "is_tangent() (geomstats.geometry.base.immersedset method)": [[3, "geomstats.geometry.base.ImmersedSet.is_tangent"]], "is_tangent() (geomstats.geometry.base.levelset method)": [[3, "geomstats.geometry.base.LevelSet.is_tangent"]], "is_tangent() (geomstats.geometry.base.openset method)": [[3, "geomstats.geometry.base.OpenSet.is_tangent"]], "is_tangent() (geomstats.geometry.base.vectorspace method)": [[3, "geomstats.geometry.base.VectorSpace.is_tangent"]], "is_tangent() (geomstats.geometry.base.vectorspaceopenset method)": [[3, "geomstats.geometry.base.VectorSpaceOpenSet.is_tangent"]], "is_tangent() (geomstats.geometry.complex_manifold.complexmanifold method)": [[3, "geomstats.geometry.complex_manifold.ComplexManifold.is_tangent"]], "is_tangent() (geomstats.geometry.discrete_surfaces.discretesurfaces method)": [[3, "geomstats.geometry.discrete_surfaces.DiscreteSurfaces.is_tangent"]], "is_tangent() (geomstats.geometry.functions.hilbertsphere method)": [[3, "geomstats.geometry.functions.HilbertSphere.is_tangent"]], "is_tangent() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.is_tangent"]], "is_tangent() (geomstats.geometry.lie_group.matrixliegroup method)": [[3, "geomstats.geometry.lie_group.MatrixLieGroup.is_tangent"]], "is_tangent() (geomstats.geometry.manifold.manifold method)": [[3, "geomstats.geometry.manifold.Manifold.is_tangent"]], "is_tangent() (geomstats.geometry.nfold_manifold.nfoldmanifold method)": [[3, "geomstats.geometry.nfold_manifold.NFoldManifold.is_tangent"]], "is_tangent() (geomstats.geometry.positive_lower_triangular_matrices.unitnormedrowspltmatrices method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.UnitNormedRowsPLTMatrices.is_tangent"]], "is_tangent() (geomstats.geometry.product_manifold.productmanifold method)": [[3, "geomstats.geometry.product_manifold.ProductManifold.is_tangent"]], "is_tangent() (geomstats.geometry.rank_k_psd_matrices.rankkpsdmatrices method)": [[3, "geomstats.geometry.rank_k_psd_matrices.RankKPSDMatrices.is_tangent"]], "is_tangent() (geomstats.geometry.sasaki_metric.tangentbundle method)": [[3, "geomstats.geometry.sasaki_metric.TangentBundle.is_tangent"]], "is_upper_triangular() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_upper_triangular"]], "is_vertical() (geomstats.geometry.fiber_bundle.fiberbundle method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.is_vertical"]], "isometry() (geomstats.geometry.siegel.siegelmetric static method)": [[3, "geomstats.geometry.siegel.SiegelMetric.isometry"]], "iterate() (geomstats.geometry.sub_riemannian_metric.subriemannianmetric static method)": [[3, "geomstats.geometry.sub_riemannian_metric.SubRiemannianMetric.iterate"]], "iterated_integrability_tensor_derivative_parallel() (geomstats.geometry.pre_shape.preshapespacebundle method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpaceBundle.iterated_integrability_tensor_derivative_parallel"]], "jacobian_diffeomorphism() (geomstats.geometry.diffeo.autodiffdiffeo method)": [[3, "geomstats.geometry.diffeo.AutodiffDiffeo.jacobian_diffeomorphism"]], "jacobian_immersion() (geomstats.geometry.base.immersedset method)": [[3, "geomstats.geometry.base.ImmersedSet.jacobian_immersion"]], "jacobian_translation() (geomstats.geometry.heisenberg.heisenbergvectors method)": [[3, "geomstats.geometry.heisenberg.HeisenbergVectors.jacobian_translation"]], "jacobian_translation() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.jacobian_translation"]], "k_landmarks (geomstats.geometry.landmarks.landmarks property)": [[3, "geomstats.geometry.landmarks.Landmarks.k_landmarks"]], "k_sampling_points (geomstats.geometry.discrete_curves.discretecurvesstartingatorigin property)": [[3, "geomstats.geometry.discrete_curves.DiscreteCurvesStartingAtOrigin.k_sampling_points"]], "ladder_parallel_transport() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.ladder_parallel_transport"]], "laplacian() (geomstats.geometry.discrete_surfaces.discretesurfaces method)": [[3, "geomstats.geometry.discrete_surfaces.DiscreteSurfaces.laplacian"]], "length() (geomstats.geometry.discrete_curves.discretecurvesstartingatorigin method)": [[3, "geomstats.geometry.discrete_curves.DiscreteCurvesStartingAtOrigin.length"]], "lie_algebra (geomstats.geometry.lie_group.liegroup attribute)": [[3, "geomstats.geometry.lie_group.LieGroup.lie_algebra"]], "lie_bracket() (geomstats.geometry.heisenberg.heisenbergvectors method)": [[3, "geomstats.geometry.heisenberg.HeisenbergVectors.lie_bracket"]], "lie_bracket() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.lie_bracket"]], "lie_bracket() (geomstats.geometry.lie_group.matrixliegroup method)": [[3, "geomstats.geometry.lie_group.MatrixLieGroup.lie_bracket"]], "lift() (geomstats.geometry.fiber_bundle.fiberbundle static method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.lift"]], "lift() (geomstats.geometry.rank_k_psd_matrices.bureswassersteinbundle method)": [[3, "geomstats.geometry.rank_k_psd_matrices.BuresWassersteinBundle.lift"]], "lifting() (geomstats.geometry.stiefel.stiefelcanonicalmetric method)": [[3, "geomstats.geometry.stiefel.StiefelCanonicalMetric.lifting"]], "log() (geomstats.geometry.complex_poincare_disk.complexpoincarediskmetric method)": [[3, "geomstats.geometry.complex_poincare_disk.ComplexPoincareDiskMetric.log"]], "log() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.log"]], "log() (geomstats.geometry.discrete_surfaces.elasticmetric method)": [[3, "geomstats.geometry.discrete_surfaces.ElasticMetric.log"]], "log() (geomstats.geometry.euclidean.euclideanmetric method)": [[3, "geomstats.geometry.euclidean.EuclideanMetric.log"]], "log() (geomstats.geometry.functions.hilbertspheremetric method)": [[3, "geomstats.geometry.functions.HilbertSphereMetric.log"]], "log() (geomstats.geometry.grassmannian.grassmanniancanonicalmetric method)": [[3, "geomstats.geometry.grassmannian.GrassmannianCanonicalMetric.log"]], "log() (geomstats.geometry.hermitian.hermitianmetric static method)": [[3, "geomstats.geometry.hermitian.HermitianMetric.log"]], "log() (geomstats.geometry.hpd_matrices.hpdaffinemetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDAffineMetric.log"]], "log() (geomstats.geometry.hpd_matrices.hpdbureswassersteinmetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDBuresWassersteinMetric.log"]], "log() (geomstats.geometry.hyperboloid.hyperboloidmetric method)": [[3, "geomstats.geometry.hyperboloid.HyperboloidMetric.log"]], "log() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, "geomstats.geometry.hypersphere.HypersphereMetric.log"]], "log() (geomstats.geometry.invariant_metric.biinvariantmetric method)": [[3, "geomstats.geometry.invariant_metric.BiInvariantMetric.log"]], "log() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.log"]], "log() (geomstats.geometry.lie_group.matrixliegroup class method)": [[3, "geomstats.geometry.lie_group.MatrixLieGroup.log"]], "log() (geomstats.geometry.minkowski.minkowskimetric method)": [[3, "geomstats.geometry.minkowski.MinkowskiMetric.log"]], "log() (geomstats.geometry.nfold_manifold.nfoldmetric method)": [[3, "geomstats.geometry.nfold_manifold.NFoldMetric.log"]], "log() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.log"]], "log() (geomstats.geometry.poincare_half_space.poincarehalfspacemetric method)": [[3, "geomstats.geometry.poincare_half_space.PoincareHalfSpaceMetric.log"]], "log() (geomstats.geometry.positive_lower_triangular_matrices.choleskymetric method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.CholeskyMetric.log"]], "log() (geomstats.geometry.positive_reals.positiverealsmetric static method)": [[3, "geomstats.geometry.positive_reals.PositiveRealsMetric.log"]], "log() (geomstats.geometry.pre_shape.preshapemetric method)": [[3, "geomstats.geometry.pre_shape.PreShapeMetric.log"]], "log() (geomstats.geometry.product_manifold.productriemannianmetric method)": [[3, "geomstats.geometry.product_manifold.ProductRiemannianMetric.log"]], "log() (geomstats.geometry.pullback_metric.pullbackdiffeometric method)": [[3, "geomstats.geometry.pullback_metric.PullbackDiffeoMetric.log"]], "log() (geomstats.geometry.quotient_metric.quotientmetric method)": [[3, "geomstats.geometry.quotient_metric.QuotientMetric.log"]], "log() (geomstats.geometry.sasaki_metric.sasakimetric method)": [[3, "geomstats.geometry.sasaki_metric.SasakiMetric.log"]], "log() (geomstats.geometry.siegel.siegelmetric method)": [[3, "geomstats.geometry.siegel.SiegelMetric.log"]], "log() (geomstats.geometry.spd_matrices.spdaffinemetric method)": [[3, "geomstats.geometry.spd_matrices.SPDAffineMetric.log"]], "log() (geomstats.geometry.spd_matrices.spdbureswassersteinmetric method)": [[3, "geomstats.geometry.spd_matrices.SPDBuresWassersteinMetric.log"]], "log() (geomstats.geometry.special_euclidean.specialeuclideanmatricescanonicalleftmetric method)": [[3, "geomstats.geometry.special_euclidean.SpecialEuclideanMatricesCanonicalLeftMetric.log"]], "log() (geomstats.geometry.stiefel.stiefelcanonicalmetric method)": [[3, "geomstats.geometry.stiefel.StiefelCanonicalMetric.log"]], "log_at_zero() (geomstats.geometry.siegel.siegelmetric static method)": [[3, "geomstats.geometry.siegel.SiegelMetric.log_at_zero"]], "log_from_identity() (geomstats.geometry.heisenberg.heisenbergvectors method)": [[3, "geomstats.geometry.heisenberg.HeisenbergVectors.log_from_identity"]], "log_from_identity() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.log_from_identity"]], "log_not_from_identity() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.log_not_from_identity"]], "logmh() (in module geomstats.geometry.spd_matrices)": [[3, "geomstats.geometry.spd_matrices.logmh"]], "matrix_representation() (geomstats.geometry.base.matrixvectorspace method)": [[3, "geomstats.geometry.base.MatrixVectorSpace.matrix_representation"]], "matrix_representation() (geomstats.geometry.general_linear.squarematrices method)": [[3, "geomstats.geometry.general_linear.SquareMatrices.matrix_representation"]], "matrix_representation() (geomstats.geometry.symmetric_matrices.symmetricmatrices static method)": [[3, "geomstats.geometry.symmetric_matrices.SymmetricMatrices.matrix_representation"]], "mean_curvature_vector() (geomstats.geometry.pullback_metric.pullbackmetric method)": [[3, "geomstats.geometry.pullback_metric.PullbackMetric.mean_curvature_vector"]], "metric_matrix() (geomstats.geometry.complex_poincare_disk.complexpoincarediskmetric method)": [[3, "geomstats.geometry.complex_poincare_disk.ComplexPoincareDiskMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.euclidean.euclideanmetric method)": [[3, "geomstats.geometry.euclidean.EuclideanMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.hermitian.hermitianmetric method)": [[3, "geomstats.geometry.hermitian.HermitianMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.hyperboloid.hyperboloidmetric method)": [[3, "geomstats.geometry.hyperboloid.HyperboloidMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, "geomstats.geometry.hypersphere.HypersphereMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.minkowski.minkowskimetric method)": [[3, "geomstats.geometry.minkowski.MinkowskiMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.nfold_manifold.nfoldmetric method)": [[3, "geomstats.geometry.nfold_manifold.NFoldMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.positive_reals.positiverealsmetric method)": [[3, "geomstats.geometry.positive_reals.PositiveRealsMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.product_manifold.productriemannianmetric method)": [[3, "geomstats.geometry.product_manifold.ProductRiemannianMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.pullback_metric.pullbackdiffeometric method)": [[3, "geomstats.geometry.pullback_metric.PullbackDiffeoMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.pullback_metric.pullbackmetric method)": [[3, "geomstats.geometry.pullback_metric.PullbackMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.metric_matrix"]], "minimize() (geomstats.geometry.sasaki_metric.gradientdescent method)": [[3, "geomstats.geometry.sasaki_metric.GradientDescent.minimize"]], "mobius_add() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.mobius_add"]], "mul() (geomstats.geometry.matrices.matrices static method)": [[3, "geomstats.geometry.matrices.Matrices.mul"]], "new() (geomstats.geometry.discrete_curves.discretecurvesstartingatorigin method)": [[3, "geomstats.geometry.discrete_curves.DiscreteCurvesStartingAtOrigin.new"]], "new() (geomstats.geometry.pre_shape.preshapespace method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpace.new"]], "norm() (geomstats.geometry.complex_matrices.complexmatricesmetric method)": [[3, "geomstats.geometry.complex_matrices.ComplexMatricesMetric.norm"]], "norm() (geomstats.geometry.euclidean.euclideanmetric method)": [[3, "geomstats.geometry.euclidean.EuclideanMetric.norm"]], "norm() (geomstats.geometry.hermitian.hermitianmetric static method)": [[3, "geomstats.geometry.hermitian.HermitianMetric.norm"]], "norm() (geomstats.geometry.matrices.matricesmetric method)": [[3, "geomstats.geometry.matrices.MatricesMetric.norm"]], "norm() (geomstats.geometry.pullback_metric.pullbackdiffeometric method)": [[3, "geomstats.geometry.pullback_metric.PullbackDiffeoMetric.norm"]], "norm() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.norm"]], "norm_factor_gradient() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, "geomstats.geometry.hypersphere.HypersphereMetric.norm_factor_gradient"]], "norm_factor_gradient() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.norm_factor_gradient"]], "normal_basis() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.normal_basis"]], "normalization_factor() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, "geomstats.geometry.hypersphere.HypersphereMetric.normalization_factor"]], "normalization_factor() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.normalization_factor"]], "normalize() (geomstats.geometry.discrete_curves.discretecurvesstartingatorigin method)": [[3, "geomstats.geometry.discrete_curves.DiscreteCurvesStartingAtOrigin.normalize"]], "normalize() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.normalize"]], "normals() (geomstats.geometry.discrete_surfaces.discretesurfaces method)": [[3, "geomstats.geometry.discrete_surfaces.DiscreteSurfaces.normals"]], "orbit() (geomstats.geometry.general_linear.generallinear class method)": [[3, "geomstats.geometry.general_linear.GeneralLinear.orbit"]], "parallel_transport() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.parallel_transport"]], "parallel_transport() (geomstats.geometry.euclidean.euclideanmetric method)": [[3, "geomstats.geometry.euclidean.EuclideanMetric.parallel_transport"]], "parallel_transport() (geomstats.geometry.grassmannian.grassmanniancanonicalmetric method)": [[3, "geomstats.geometry.grassmannian.GrassmannianCanonicalMetric.parallel_transport"]], "parallel_transport() (geomstats.geometry.hpd_matrices.hpdaffinemetric method)": [[3, 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(geomstats.geometry.discrete_surfaces.elasticmetric method)": [[3, "geomstats.geometry.discrete_surfaces.ElasticMetric.path_energy_per_time"]], "point_ndim (geomstats.geometry.manifold.manifold attribute)": [[3, "geomstats.geometry.manifold.Manifold.point_ndim"]], "pointwise_inner_product() (geomstats.geometry.nfold_manifold.nfoldmetric method)": [[3, "geomstats.geometry.nfold_manifold.NFoldMetric.pointwise_inner_product"]], "pointwise_norm() (geomstats.geometry.nfold_manifold.nfoldmetric method)": [[3, "geomstats.geometry.nfold_manifold.NFoldMetric.pointwise_norm"]], "powermh() (in module geomstats.geometry.hermitian_matrices)": [[3, "geomstats.geometry.hermitian_matrices.powermh"]], "projection() (geomstats.geometry.base.complexvectorspace static method)": [[3, "geomstats.geometry.base.ComplexVectorSpace.projection"]], "projection() (geomstats.geometry.base.complexvectorspaceopenset method)": [[3, "geomstats.geometry.base.ComplexVectorSpaceOpenSet.projection"]], "projection() 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(geomstats.geometry.functions.hilbertsphere method)": [[3, "geomstats.geometry.functions.HilbertSphere.random_point"]], "random_point() (geomstats.geometry.general_linear.generallinear method)": [[3, "geomstats.geometry.general_linear.GeneralLinear.random_point"]], "random_point() (geomstats.geometry.grassmannian.grassmannian method)": [[3, "geomstats.geometry.grassmannian.Grassmannian.random_point"]], "random_point() (geomstats.geometry.hermitian_matrices.hermitianmatrices method)": [[3, "geomstats.geometry.hermitian_matrices.HermitianMatrices.random_point"]], "random_point() (geomstats.geometry.hpd_matrices.hpdmatrices method)": [[3, "geomstats.geometry.hpd_matrices.HPDMatrices.random_point"]], "random_point() (geomstats.geometry.manifold.manifold method)": [[3, "geomstats.geometry.manifold.Manifold.random_point"]], "random_point() (geomstats.geometry.matrices.matrices method)": [[3, "geomstats.geometry.matrices.Matrices.random_point"]], "random_point() (geomstats.geometry.nfold_manifold.nfoldmanifold method)": [[3, "geomstats.geometry.nfold_manifold.NFoldManifold.random_point"]], "random_point() (geomstats.geometry.positive_lower_triangular_matrices.positivelowertriangularmatrices method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.PositiveLowerTriangularMatrices.random_point"]], "random_point() (geomstats.geometry.positive_reals.positivereals static method)": [[3, "geomstats.geometry.positive_reals.PositiveReals.random_point"]], "random_point() (geomstats.geometry.pre_shape.preshapespace method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpace.random_point"]], "random_point() (geomstats.geometry.product_manifold.productmanifold method)": [[3, "geomstats.geometry.product_manifold.ProductManifold.random_point"]], "random_point() (geomstats.geometry.rank_k_psd_matrices.rankkpsdmatrices method)": [[3, "geomstats.geometry.rank_k_psd_matrices.RankKPSDMatrices.random_point"]], "random_point() (geomstats.geometry.sasaki_metric.tangentbundle method)": [[3, "geomstats.geometry.sasaki_metric.TangentBundle.random_point"]], "random_point() (geomstats.geometry.siegel.siegel method)": [[3, "geomstats.geometry.siegel.Siegel.random_point"]], "random_point() (geomstats.geometry.skew_symmetric_matrices.skewsymmetricmatrices method)": [[3, "geomstats.geometry.skew_symmetric_matrices.SkewSymmetricMatrices.random_point"]], "random_point() (geomstats.geometry.spd_matrices.spdmatrices method)": [[3, "geomstats.geometry.spd_matrices.SPDMatrices.random_point"]], "random_point() (geomstats.geometry.special_euclidean.specialeuclideanmatricesliealgebra method)": [[3, "geomstats.geometry.special_euclidean.SpecialEuclideanMatricesLieAlgebra.random_point"]], "random_point() (geomstats.geometry.stiefel.stiefel method)": [[3, "geomstats.geometry.stiefel.Stiefel.random_point"]], "random_tangent_vec() (geomstats.geometry.base.diffeomorphicmanifold method)": [[3, "geomstats.geometry.base.DiffeomorphicManifold.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.complex_manifold.complexmanifold method)": [[3, "geomstats.geometry.complex_manifold.ComplexManifold.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.hpd_matrices.hpdmatrices method)": [[3, "geomstats.geometry.hpd_matrices.HPDMatrices.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.manifold.manifold method)": [[3, "geomstats.geometry.manifold.Manifold.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.product_manifold.productmanifold method)": [[3, "geomstats.geometry.product_manifold.ProductManifold.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.sasaki_metric.tangentbundle method)": [[3, "geomstats.geometry.sasaki_metric.TangentBundle.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.siegel.siegel method)": [[3, "geomstats.geometry.siegel.Siegel.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.spd_matrices.spdmatrices method)": [[3, "geomstats.geometry.spd_matrices.SPDMatrices.random_tangent_vec"]], "random_uniform() (geomstats.geometry.grassmannian.grassmannian method)": [[3, "geomstats.geometry.grassmannian.Grassmannian.random_uniform"]], "random_uniform() (geomstats.geometry.pre_shape.preshapespace method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpace.random_uniform"]], "random_uniform() (geomstats.geometry.stiefel.stiefel method)": [[3, "geomstats.geometry.stiefel.Stiefel.random_uniform"]], "random_unit_tangent_vec() (geomstats.geometry.complex_riemannian_metric.complexriemannianmetric method)": [[3, "geomstats.geometry.complex_riemannian_metric.ComplexRiemannianMetric.random_unit_tangent_vec"]], "random_unit_tangent_vec() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.random_unit_tangent_vec"]], "regularize() (geomstats.geometry.base.diffeomorphicmanifold method)": [[3, "geomstats.geometry.base.DiffeomorphicManifold.regularize"]], "regularize() (geomstats.geometry.complex_manifold.complexmanifold method)": [[3, "geomstats.geometry.complex_manifold.ComplexManifold.regularize"]], "regularize() (geomstats.geometry.hyperboloid.hyperboloid method)": [[3, "geomstats.geometry.hyperboloid.Hyperboloid.regularize"]], "regularize() (geomstats.geometry.manifold.manifold method)": [[3, "geomstats.geometry.manifold.Manifold.regularize"]], "regularize() (geomstats.geometry.product_manifold.productmanifold method)": [[3, "geomstats.geometry.product_manifold.ProductManifold.regularize"]], "reshape() (geomstats.geometry.matrices.matrices method)": [[3, "geomstats.geometry.matrices.Matrices.reshape"]], "retraction() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.retraction"]], "retraction() (geomstats.geometry.stiefel.stiefelcanonicalmetric static method)": [[3, "geomstats.geometry.stiefel.StiefelCanonicalMetric.retraction"]], "ricci_tensor() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.ricci_tensor"]], "riemann_sum() (geomstats.geometry.discrete_curves.l2curvesmetric static method)": [[3, "geomstats.geometry.discrete_curves.L2CurvesMetric.riemann_sum"]], "riemann_tensor() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.riemann_tensor"]], "riemannian_submersion() (geomstats.geometry.fiber_bundle.fiberbundle static method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.riemannian_submersion"]], "riemannian_submersion() (geomstats.geometry.full_rank_correlation_matrices.correlationmatricesbundle static method)": [[3, "geomstats.geometry.full_rank_correlation_matrices.CorrelationMatricesBundle.riemannian_submersion"]], "riemannian_submersion() (geomstats.geometry.rank_k_psd_matrices.bureswassersteinbundle static method)": [[3, "geomstats.geometry.rank_k_psd_matrices.BuresWassersteinBundle.riemannian_submersion"]], "scalar_curvature() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.scalar_curvature"]], "second_centered_difference() (in module geomstats.geometry.discrete_curves)": [[3, "geomstats.geometry.discrete_curves.second_centered_difference"]], "second_fundamental_form() (geomstats.geometry.pullback_metric.pullbackmetric method)": [[3, "geomstats.geometry.pullback_metric.PullbackMetric.second_fundamental_form"]], "sectional_curvature() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.sectional_curvature"]], "sectional_curvature() (geomstats.geometry.siegel.siegelmetric method)": [[3, "geomstats.geometry.siegel.SiegelMetric.sectional_curvature"]], "sectional_curvature_at_zero() (geomstats.geometry.siegel.siegelmetric method)": [[3, "geomstats.geometry.siegel.SiegelMetric.sectional_curvature_at_zero"]], "shape (geomstats.geometry.product_manifold.productriemannianmetric property)": [[3, "geomstats.geometry.product_manifold.ProductRiemannianMetric.shape"]], "squared_dist() (geomstats.geometry.complex_poincare_disk.complexpoincarediskmetric method)": [[3, "geomstats.geometry.complex_poincare_disk.ComplexPoincareDiskMetric.squared_dist"]], "squared_dist() (geomstats.geometry.grassmannian.grassmanniancanonicalmetric method)": [[3, "geomstats.geometry.grassmannian.GrassmannianCanonicalMetric.squared_dist"]], "squared_dist() (geomstats.geometry.hpd_matrices.hpdbureswassersteinmetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDBuresWassersteinMetric.squared_dist"]], "squared_dist() (geomstats.geometry.hyperboloid.hyperboloidmetric method)": [[3, "geomstats.geometry.hyperboloid.HyperboloidMetric.squared_dist"]], "squared_dist() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, 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\ No newline at end of file
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"geomstats-test-cases-geometry-stratified-package"]], "geomstats.test_cases.geometry.stratified.point_set module": [[14, "geomstats-test-cases-geometry-stratified-point-set-module"]], "geomstats.test_cases.information_geometry package": [[15, "geomstats-test-cases-information-geometry-package"]], "geomstats.test_cases.information_geometry.base module": [[15, "geomstats-test-cases-information-geometry-base-module"]], "geomstats.test_cases.information_geometry.beta module": [[15, "geomstats-test-cases-information-geometry-beta-module"]], "geomstats.test_cases.information_geometry.binomial module": [[15, "geomstats-test-cases-information-geometry-binomial-module"]], "geomstats.test_cases.information_geometry.dirichlet module": [[15, "geomstats-test-cases-information-geometry-dirichlet-module"]], "geomstats.test_cases.information_geometry.exponential module": [[15, "geomstats-test-cases-information-geometry-exponential-module"]], "geomstats.test_cases.information_geometry.gamma module": [[15, "geomstats-test-cases-information-geometry-gamma-module"]], "geomstats.test_cases.information_geometry.geometric module": [[15, "geomstats-test-cases-information-geometry-geometric-module"]], "geomstats.test_cases.information_geometry.multinomial module": [[15, "geomstats-test-cases-information-geometry-multinomial-module"]], "geomstats.test_cases.information_geometry.normal module": [[15, "geomstats-test-cases-information-geometry-normal-module"]], "geomstats.test_cases.information_geometry.poisson module": [[15, "geomstats-test-cases-information-geometry-poisson-module"]], "geomstats.test_cases.learning package": [[16, "geomstats-test-cases-learning-package"]], "geomstats.test_cases.learning.agglomerative_hierarchical_clustering module": [[16, "geomstats-test-cases-learning-agglomerative-hierarchical-clustering-module"]], "geomstats.test_cases.learning.expectation_maximization module": [[16, "geomstats-test-cases-learning-expectation-maximization-module"]], "geomstats.test_cases.learning.exponential_barycenter module": [[16, "module-geomstats.test_cases.learning.exponential_barycenter"]], "geomstats.test_cases.learning.frechet_mean module": [[16, "geomstats-test-cases-learning-frechet-mean-module"]], "geomstats.test_cases.learning.geodesic_regression module": [[16, "geomstats-test-cases-learning-geodesic-regression-module"]], "geomstats.test_cases.learning.incremental_frechet_mean module": [[16, "geomstats-test-cases-learning-incremental-frechet-mean-module"]], "geomstats.test_cases.learning.kalman_filter module": [[16, "geomstats-test-cases-learning-kalman-filter-module"]], "geomstats.test_cases.learning.kernel_density_estimation_classifier module": [[16, "geomstats-test-cases-learning-kernel-density-estimation-classifier-module"]], "geomstats.test_cases.learning.kmeans module": [[16, "geomstats-test-cases-learning-kmeans-module"]], "geomstats.test_cases.learning.knn module": [[16, "geomstats-test-cases-learning-knn-module"]], "geomstats.test_cases.learning.mdm module": [[16, "geomstats-test-cases-learning-mdm-module"]], "geomstats.test_cases.learning.pca module": [[16, "geomstats-test-cases-learning-pca-module"]], "geomstats.test_cases.learning.preprocessing module": [[16, "geomstats-test-cases-learning-preprocessing-module"]], "geomstats.test_cases.learning.radial_kernel_functions module": [[16, "geomstats-test-cases-learning-radial-kernel-functions-module"]], "geomstats.test_cases.learning.wrapped_gaussian_process module": [[16, "geomstats-test-cases-learning-wrapped-gaussian-process-module"]], "geomstats.test_cases.numerics package": [[17, "geomstats-test-cases-numerics-package"]], "geomstats.test_cases.numerics.geodesic module": [[17, "geomstats-test-cases-numerics-geodesic-module"]], "geomstats.visualization package": [[18, "geomstats-visualization-package"]], "geomstats.visualization.hyperbolic module": [[18, "module-geomstats.visualization.hyperbolic"]], "geomstats.visualization.hypersphere module": [[18, "module-geomstats.visualization.hypersphere"]], "geomstats.visualization.poincare_polydisk module": [[18, "module-geomstats.visualization.poincare_polydisk"]], "geomstats.visualization.pre_shape module": [[18, "module-geomstats.visualization.pre_shape"]], "geomstats.visualization.spd_matrices module": [[18, "module-geomstats.visualization.spd_matrices"]], "geomstats.visualization.special_euclidean module": [[18, "module-geomstats.visualization.special_euclidean"]], "geomstats.visualization.special_orthogonal module": [[18, "module-geomstats.visualization.special_orthogonal"]], "API Reference": [[19, "api-reference"]], "Packages & Modules": [[19, null]], "geomstats": [[20, "geomstats"]], "Contributing Guide": [[21, "contributing-guide"]], "Development Setup": [[21, "development-setup"]], "Source control with Git": [[21, "source-control-with-git"]], "Getting the code": [[21, "getting-the-code"]], "Dependencies and a virtual environment": [[21, "dependencies-and-a-virtual-environment"]], "Backends": [[21, "backends"]], "Run the tests": [[21, "run-the-tests"]], "Build the docs": [[21, "build-the-docs"]], "Folder Structure": [[21, "folder-structure"]], "Testing": [[21, "testing"]], "Test Driven Development": [[21, "test-driven-development"]], "TDD with pytest": [[21, "tdd-with-pytest"]], "Writing tests for geomstats": [[21, "writing-tests-for-geomstats"]], "Running tests": [[21, "running-tests"]], "Workflow of a contribution": [[21, "workflow-of-a-contribution"]], "Create or choose an issue for new contributors": [[21, "create-or-choose-an-issue-for-new-contributors"]], "good first issue tag": [[21, null]], "starter project tag": [[21, null]], "help wanted tag": [[21, null]], "Making changes": [[21, "making-changes"]], "Pull Request Checklist": [[21, "pull-request-checklist"]], "Stalled Pull Requests": [[21, "stalled-pull-requests"]], "Coding Style Guidelines": [[21, "coding-style-guidelines"]], "Documentation": [[21, "documentation"]], "Building the Documentation": [[21, "building-the-documentation"]], "Writing Docstrings": [[21, "writing-docstrings"]], "Intro to Docstrings": [[21, "intro-to-docstrings"]], "The Anatomy of a Docstring": [[21, "the-anatomy-of-a-docstring"]], "Docstring Examples": [[21, "docstring-examples"]], "Code Review Guidelines": [[21, "code-review-guidelines"]], "Reporting bugs and features": [[21, "reporting-bugs-and-features"]], "Issue tracker": [[21, "issue-tracker"]], "Template of a bug/issue report": [[21, "template-of-a-bug-issue-report"]], "Issue Triaging": [[21, "issue-triaging"]], "Reproducing issues": [[21, "reproducing-issues"]], "Commenting on alternative solutions": [[21, "commenting-on-alternative-solutions"]], "Answering questions": [[21, "answering-questions"]], "Labelling and assigning the issue": [[21, "labelling-and-assigning-the-issue"]], "Explanation": [[22, "explanation"]], "Code Examples": [[23, "code-examples"]], "Learn differential geometry": [[23, "learn-differential-geometry"]], "Run learning algorithms": [[23, "run-learning-algorithms"]], "First steps": [[24, "first-steps"]], "Getting Started": [[25, "getting-started"]], "Geomstats governance and decision-making": [[26, "geomstats-governance-and-decision-making"]], "Roles And Responsibilities": [[26, "roles-and-responsibilities"]], "Contributors": [[26, "contributors"]], "Contributor Experience Team": [[26, "contributor-experience-team"]], "Communication team": [[26, "communication-team"]], "Core developers": [[26, "core-developers"]], "Technical Committee": [[26, "technical-committee"]], "Decision Making Process": [[26, "decision-making-process"]], "Conflict of Interest": [[26, "conflict-of-interest"]], "Breach": [[26, "breach"]], "Changing the Governance Documents": [[26, "changing-the-governance-documents"]], "Google Season of Docs: Case Study": [[27, "google-season-of-docs-case-study"]], "Design and Organize Scattered Documentation in Geomstats": [[27, "design-and-organize-scattered-documentation-in-geomstats"]], "Problem Statement": [[27, "problem-statement"]], "Proposal Abstract": [[27, "proposal-abstract"]], "Project Description": [[27, "project-description"]], "Budget": [[27, "budget"]], "Participants": [[27, "participants"]], "Timeline and Results": [[27, "timeline-and-results"]], "Title": [[27, "id4"]], "Metrics": [[27, "metrics"]], "Analysis": [[27, "analysis"]], "Summary": [[27, "summary"]], "Google Season of Docs: Project Page": [[27, "google-season-of-docs-project-page"]], "About the organization": [[27, "about-the-organization"]], "About the project": [[27, "about-the-project"]], "The problem": [[27, "the-problem"]], "The project\u2019s scope": [[27, "the-projects-scope"]], "Measuring project\u2019s success": [[27, "measuring-projects-success"]], "Proposed Budget": [[27, "proposed-budget"]], "Additional information": [[27, "additional-information"]], "Important links for technical writers": [[27, "important-links-for-technical-writers"]], "Hackathons": [[28, "hackathons"]], "Workflow": [[28, "workflow"]], "Past and future hackathons": [[28, "past-and-future-hackathons"]], "Related events": [[28, "related-events"]], "Hackathon at Institut Henri Poincar\u00e9, 17th-21st Oct 2022": [[29, "hackathon-at-institut-henri-poincare-17th-21st-oct-2022"]], "Future changes": [[29, "future-changes"]], "Equip with metric": [[29, "equip-with-metric"]], "Intrinsic vs extrinsic coordinates (manifold)": [[29, "intrinsic-vs-extrinsic-coordinates-manifold"]], "Intrinsic vs extrinsic coordinates (connection)": [[29, "intrinsic-vs-extrinsic-coordinates-connection"]], "No dynamic behavior": [[29, "no-dynamic-behavior"]], "numerics": [[29, "numerics"]], "gs.matrices": [[29, "gs-matrices"]], "Points shape": [[29, "points-shape"]], "New structure": [[29, "new-structure"]], "Geomstats": [[30, "geomstats"]], "Objectives": [[30, "objectives"]], "Design": [[30, "design"]], "Learn More": [[30, "learn-more"]], "What is the motivation for analyzing data on manifolds?": [[31, "What-is-the-motivation-for-analyzing-data-on-manifolds?"]], "1) Analyzing data on a manifold reduces the degrees of freedom of the system, making computations less complicated.": [[31, "1)-Analyzing-data-on-a-manifold-reduces-the-degrees-of-freedom-of-the-system,-making-computations-less-complicated."]], "2) Knowing the manifold that a data set belongs to may give you a deeper understanding of your data\u2019s evolution or \u201cwhy your data looks/evolves the way it does\u201d": [[31, "2)-Knowing-the-manifold-that-a-data-set-belongs-to-may-give-you-a-deeper-understanding-of-your-data's-evolution-or-%22why-your-data-looks/evolves-the-way-it-does%22"]], "3) Knowing the manifold a data set lives in will help you give you better predictive power and extract the \u201csignal\u201d from a noisy data set or a data set with very few datapoints.": [[31, "3)-Knowing-the-manifold-a-data-set-lives-in-will-help-you-give-you-better-predictive-power-and-extract-the-%22signal%22-from-a-noisy-data-set-or-a-data-set-with-very-few-datapoints."]], "What will you learn in these tutorials?": [[31, "What-will-you-learn-in-these-tutorials?"]], "1. Introduction": [[32, "1.-Introduction"]], "2. What is a Manifold?": [[32, "2.-What-is-a-Manifold?"]], "2.1 Precise Mathematical Definition": [[32, "2.1-Precise-Mathematical-Definition"]], "2.2 Imprecise \u201cLayman\u2019s Terms\u201d Definition:": [[32, "2.2-Imprecise-%22Layman's-Terms%22-Definition:"]], "2.3 Hypersphere example:": [[32, "2.3-Hypersphere-example:"]], "2.4 More examples of manifolds:": [[32, "2.4-More-examples-of-manifolds:"]], "3. Why do we care about manifolds?": [[32, "3.-Why-do-we-care-about-manifolds?"]], "4. What does the Manifold class do?": [[32, "4.-What-does-the-Manifold-class-do?"]], "4.1 Tangent Vectors": [[32, "4.1-Tangent-Vectors"]], "4.2 Tangent Spaces": [[32, "4.2-Tangent-Spaces"]], "5. How is the Manifold class structured?": [[32, "5.-How-is-the-Manifold-class-structured?"]], "5.1 The Parent Class: Manifold": [[32, "5.1-The-Parent-Class:-Manifold"]], "5.1.1 Examples of Using Manifold\u2019s Attributes and Methods in the Subclass: Hypersphere": [[32, "5.1.1-Examples-of-Using-Manifold's-Attributes-and-Methods-in-the-Subclass:-Hypersphere"]], "5.1.2 The Full Manifold Class Code": [[32, "5.1.2-The-Full-Manifold-Class-Code"]], "5.2 VectorSpaceOpenSet": [[32, "5.2-VectorSpaceOpenSet"]], "5.2.1 What is an Open Set?": [[32, "5.2.1-What-is-an-Open-Set?"]], "5.2.2 What Methods are Implemented in VectorSpaceOpenSet?": [[32, "5.2.2-What-Methods-are-Implemented-in-VectorSpaceOpenSet?"]], "5.3 LevelSet": [[32, "5.3-LevelSet"]], "5.3.1 What is a Level Set?": [[32, "5.3.1-What-is-a-Level-Set?"]], "5.3.2 LevelSet in Geomstats": [[32, "5.3.2-LevelSet-in-Geomstats"]], "5.4 VectorSpace": [[32, "5.4-VectorSpace"]], "5.5 ProductManifold": [[32, "5.5-ProductManifold"]], "5.5.1 Example": [[32, "5.5.1-Example"]], "Conclusion": [[32, "Conclusion"], [36, "Conclusion"], [49, "Conclusion"]], "1. What is a Connection?": [[33, "1.-What-is-a-Connection?"]], "1.1 General Mathematical Definition of the Connection": [[33, "1.1-General-Mathematical-Definition-of-the-Connection"]], "1.2 What does the Connection actually do? \u2013 a more friendly description of the connection, with an example": [[33, "1.2-What-does-the-Connection-actually-do?----a-more-friendly-description-of-the-connection,-with-an-example"]], "2. Why is the Connection important (big picture)? \u2013 parallel transport, parallel vector fields": [[33, "2.-Why-is-the-Connection-important-(big-picture)?----parallel-transport,-parallel-vector-fields"]], "2.1 Parallel Vector Fields": [[33, "2.1-Parallel-Vector-Fields"]], "2.2 Parallel Transport": [[33, "2.2-Parallel-Transport"]], "3. Defining Geodesics with the Connection": [[33, "3.-Defining-Geodesics-with-the-Connection"]], "3.1 Intuitive Definition of a Geodesic": [[33, "3.1-Intuitive-Definition-of-a-Geodesic"]], "3.2 Defining a geodesic with the connection": [[33, "3.2-Defining-a-geodesic-with-the-connection"]], "3.3 Building Intuition with Physics": [[33, "3.3-Building-Intuition-with-Physics"]], "4. Description of Connection in Local Coordinates (using Christoffel Symbols)": [[33, "4.-Description-of-Connection-in-Local-Coordinates-(using-Christoffel-Symbols)"]], "4.1 Math Notation: \\mathbf{ \\partial_i} Basis Vector": [[33, "4.1-Math-Notation:-\\mathbf{-\\partial_i}-Basis-Vector"]], "4.2 Einstein Notation": [[33, "4.2-Einstein-Notation"]], "4.3 What does a Christoffel Symbol represent?": [[33, "4.3-What-does-a-Christoffel-Symbol-represent?"]], "4.4 The Geodesic Equation in local coordinates": [[33, "4.4-The-Geodesic-Equation-in-local-coordinates"]], "4.5 Why should we care about Christoffel Symbols and this representation of the connection?": [[33, "4.5-Why-should-we-care-about-Christoffel-Symbols-and-this-representation-of-the-connection?"]], "5. Additional Important Mathematical concepts utilized by Connection Class: Logarithm, Exponential": [[33, "5.-Additional-Important-Mathematical-concepts-utilized-by-Connection-Class:-Logarithm,-Exponential"]], "Exponentials on Manifolds": [[33, "Exponentials-on-Manifolds"]], "Logarithms on Manifolds": [[33, "Logarithms-on-Manifolds"]], "6. Geomstats Documentation: The Connection Class": [[33, "6.-Geomstats-Documentation:-The-Connection-Class"]], "6.1 method: parallel_transport()": [[33, "6.1-method:-parallel_transport()"]], "6.2 method: geodesic_equation()": [[33, "6.2-method:-geodesic_equation()"]], "6.3 method: exp() (exponential)": [[33, "6.3-method:-exp()-(exponential)"]], "6.4 method: log() (logarithm)": [[33, "6.4-method:-log()-(logarithm)"]], "6.5 The full Connection class": [[33, "6.5-The-full-Connection-class"]], "7. RiemannianMetric Sub-Class": [[33, "7.-RiemannianMetric-Sub-Class"]], "7.1 Riemannian Metrics": [[33, "7.1-Riemannian-Metrics"]], "7.2 How is the Riemannian Metric Connected with the Connection?": [[33, "7.2-How-is-the-Riemannian-Metric-Connected-with-the-Connection?"]], "7.3 RiemannianMetric in Geomstats": [[33, "7.3-RiemannianMetric-in-Geomstats"]], "8. Conclusion": [[33, "8.-Conclusion"]], "Data on Manifolds": [[34, "Data-on-Manifolds"]], "From data on linear spaces to data on manifolds": [[34, "From-data-on-linear-spaces-to-data-on-manifolds"]], "Examples of data on manifolds": [[34, "Examples-of-data-on-manifolds"]], "World cities: data on the sphere": [[34, "World-cities:-data-on-the-sphere"]], "Pose of objects in pictures: data on the Lie group of 3D rotations": [[34, "Pose-of-objects-in-pictures:-data-on-the-Lie-group-of-3D-rotations"]], "Social networks: data on the hyperbolic space": [[34, "Social-networks:-data-on-the-hyperbolic-space"]], "Brain connectomes: data on the manifold of Symmetric Positive Definite (SPD) matrices": [[34, "Brain-connectomes:-data-on-the-manifold-of-Symmetric-Positive-Definite-(SPD)-matrices"]], "Monkey\u2019s optical nerve heads: Data as landmarks in 3D": [[34, "Monkey's-optical-nerve-heads:-Data-as-landmarks-in-3D"]], "Reference:": [[34, "Reference:"]], "From vector spaces to manifolds": [[35, "From-vector-spaces-to-manifolds"], [35, "id1"]], "From addition to exponential map": [[35, "From-addition-to-exponential-map"]], "From substraction to logarithm map": [[35, "From-substraction-to-logarithm-map"]], "Geodesics": [[35, "Geodesics"], [47, "Geodesics"]], "Geodesics on the hyperbolic plane": [[35, "Geodesics-on-the-hyperbolic-plane"]], "Geodesics on the special euclidean group SE(3)": [[35, "Geodesics-on-the-special-euclidean-group-SE(3)"]], "Learning on Tangent Data": [[36, "Learning-on-Tangent-Data"]], "The Data": [[36, "The-Data"], [43, "The-Data"]], "The Manifold": [[36, "The-Manifold"]], "The Transformer": [[36, "The-Transformer"]], "References": [[36, "References"], [39, "References"], [41, "References"], [44, "References"], [45, "References"], [46, "References"], [47, "References"]], "Fr\u00e9chet Mean and Tangent PCA": [[37, "Fr\u00e9chet-Mean-and-Tangent-PCA"]], "On the sphere": [[37, "On-the-sphere"]], "Generate data on the sphere": [[37, "Generate-data-on-the-sphere"]], "Fr\u00e9chet mean": [[37, "Fr\u00e9chet-mean"], [37, "id1"]], "Tangent PCA (at the Fr\u00e9chet mean)": [[37, "Tangent-PCA-(at-the-Fr\u00e9chet-mean)"], [37, "id2"]], "In the Hyperbolic plane": [[37, "In-the-Hyperbolic-plane"]], "Generate data on the hyperbolic plane": [[37, "Generate-data-on-the-hyperbolic-plane"]], "K-Means clustering on a Riemannian Manifold": [[38, "K-Means-clustering-on-a-Riemannian-Manifold"]], "Data init: Generate clusters randomly on the sphere": [[38, "Data-init:-Generate-clusters-randomly-on-the-sphere"]], "Operate Riemannian K-Means clustering": [[38, "Operate-Riemannian-K-Means-clustering"]], "Information geometry": [[39, "Information-geometry"]], "Introduction": [[39, "Introduction"], [44, "Introduction"]], "Normal distributions": [[39, "Normal-distributions"]], "Beta distributions": [[39, "Beta-distributions"]], "Implement your own Riemannian Geometry": [[40, "Implement-your-own-Riemannian-Geometry"]], "Immersed manifolds: the example of the 2-sphere": [[40, "Immersed-manifolds:-the-example-of-the-2-sphere"]], "Immersed manifolds: the example of a surface defined by its graph": [[40, "Immersed-manifolds:-the-example-of-a-surface-defined-by-its-graph"]], "Pullback via diffeomorphism: Warping of the plane \\mathbb{R}^2": [[40, "Pullback-via-diffeomorphism:-Warping-of-the-plane-\\mathbb{R}^2"]], "Shape analysis of curves with the Square Root Velocity metric": [[41, "Shape-analysis-of-curves-with-the-Square-Root-Velocity-metric"]], "1. A first example in 2D": [[41, "1.-A-first-example-in-2D"]], "Remarks:": [[41, "Remarks:"]], "2. Focus on quotienting out reparametrizations": [[41, "2.-Focus-on-quotienting-out-reparametrizations"]], "2.1. Comparing parametrized curves": [[41, "2.1.-Comparing-parametrized-curves"]], "2.2. Comparing unparametrized curves": [[41, "2.2.-Comparing-unparametrized-curves"]], "3. Example in 3D": [[41, "3.-Example-in-3D"]], "Shape Analysis of Cancer Cells": [[42, "Shape-Analysis-of-Cancer-Cells"]], "1. Introduction and Motivation": [[42, "1.-Introduction-and-Motivation"], [49, "1.-Introduction-and-Motivation"]], "2. Dataset Description": [[42, "2.-Dataset-Description"]], "3. Preprocessing": [[42, "3.-Preprocessing"]], "Interpolation: Encoding Discrete Curves With Same Number of Points": [[42, "Interpolation:-Encoding-Discrete-Curves-With-Same-Number-of-Points"]], "Visualization of Interpolated Dataset of Curves": [[42, "Visualization-of-Interpolated-Dataset-of-Curves"]], "Remove duplicate samples in curves": [[42, "Remove-duplicate-samples-in-curves"]], "Alignment": [[42, "Alignment"]], "4 Data Analysis": [[42, "4-Data-Analysis"]], "Analyze Distances to the \u201cGlobal\u201d Mean Shape": [[42, "Analyze-Distances-to-the-%22Global%22-Mean-Shape"]], "Changing the Metric on the Space of Cell Shapes": [[42, "Changing-the-Metric-on-the-Space-of-Cell-Shapes"]], "Hypothesis Testing on Equality of Means": [[42, "Hypothesis-Testing-on-Equality-of-Means"]], "Visualization of the Mean of each Treatment": [[42, "Visualization-of-the-Mean-of-each-Treatment"]], "Distance of the Cell Shapes to their Own Mean Shape": [[42, "Distance-of-the-Cell-Shapes-to-their-Own-Mean-Shape"]], "Hand gesture classification with EMG data using Riemannian metrics": [[43, "Hand-gesture-classification-with-EMG-data-using-Riemannian-metrics"]], "Context": [[43, "Context"]], "Parameters": [[43, "Parameters"]], "Preprocessing into covariance matrices": [[43, "Preprocessing-into-covariance-matrices"]], "Covariances plot of the euclidean average": [[43, "Covariances-plot-of-the-euclidean-average"]], "Covariances plot of the Frechet Mean of the affine invariant metric": [[43, "Covariances-plot-of-the-Frechet-Mean-of-the-affine-invariant-metric"]], "Sign Classification": [[43, "Sign-Classification"]], "A. Test on the same session and user as Training/Calibration": [[43, "A.-Test-on-the-same-session-and-user-as-Training/Calibration"]], "A.0. Using Logistic Regression on the vectorized Matrix (Euclidean Method)": [[43, "A.0.-Using-Logistic-Regression-on-the-vectorized-Matrix-(Euclidean-Method)"]], "A.1. Using MLP on the vectorized Matrix (Euclidean Method)": [[43, "A.1.-Using-MLP-on-the-vectorized-Matrix-(Euclidean-Method)"]], "A.2. Using Tangent space projection + Logistic Regression": [[43, "A.2.-Using-Tangent-space-projection-+-Logistic-Regression"]], "A.3. Using Tangent space projection + MLP": [[43, "A.3.-Using-Tangent-space-projection-+-MLP"]], "A.4. Using Euclidean MDM": [[43, "A.4.-Using-Euclidean-MDM"]], "A.5. Using Riemannian MDM": [[43, "A.5.-Using-Riemannian-MDM"]], "Summary plots": [[43, "Summary-plots"]], "Hyperbolic Embedding of Graphs and Clustering": [[44, "Hyperbolic-Embedding-of-Graphs-and-Clustering"]], "Parameters and Initialization": [[44, "Parameters-and-Initialization"]], "Loss function.": [[44, "Loss-function."]], "Riemannian optimization.": [[44, "Riemannian-optimization."]], "Capturing the graph structure": [[44, "Capturing-the-graph-structure"]], "Numerically optimizing the loss function": [[44, "Numerically-optimizing-the-loss-function"]], "Plotting results": [[44, "Plotting-results"]], "Classifying hands poses with Kendall shape spaces": [[45, "Classifying-hands-poses-with-Kendall-shape-spaces"]], "Hands shapes": [[45, "Hands-shapes"]], "Computing with shapes of landmarks in Kendall shape spaces": [[46, "Computing-with-shapes-of-landmarks-in-Kendall-shape-spaces"]], "Computing with triangular shapes in Kendall framework": [[47, "Computing-with-triangular-shapes-in-Kendall-framework"]], "The space \\Sigma_2^3 of 2D triangles": [[47, "The-space-\\Sigma_2^3-of-2D-triangles"]], "Uniform distribution": [[47, "Uniform-distribution"], [47, "id1"]], "Geodesic": [[47, "Geodesic"]], "Parallel transport": [[47, "Parallel-transport"], [47, "id2"]], "The space \\Sigma_3^3 of 3D triangles": [[47, "The-space-\\Sigma_3^3-of-3D-triangles"]], "Usecase : optical nerve heads": [[47, "Usecase-:-optical-nerve-heads"]], "Stratified spaces": [[48, "Stratified-spaces"]], "Spider": [[48, "Spider"]], "Graph Space": [[48, "Graph-Space"], [51, "Graph-Space"]], "Graph": [[48, "Graph"]], "Set of Graphs: GraphSpace": [[48, "Set-of-Graphs:-GraphSpace"]], "Labelled or Unlabelled?": [[48, "Labelled-or-Unlabelled?"]], "Graph Space Metric": [[48, "Graph-Space-Metric"]], "Graph to Graph Alignment": [[48, "Graph-to-Graph-Alignment"]], "Graph to Geodesic Alignment": [[48, "Graph-to-Geodesic-Alignment"]], "Optimization of Sao Paulo traffic": [[49, "Optimization-of-Sao-Paulo-traffic"]], "2. Dataset description": [[49, "2.-Dataset-description"]], "Pre-processing the dataset": [[49, "Pre-processing-the-dataset"]], "3. Mathematical modeling": [[49, "3.-Mathematical-modeling"]], "3.1. Road representation: introduction to the Gamma manifold.": [[49, "3.1.-Road-representation:-introduction-to-the-Gamma-manifold."]], "3.1.1. Hypotheses": [[49, "3.1.1.-Hypotheses"]], "3.1.2. Model": [[49, "3.1.2.-Model"]], "3.2. Mathematical representation of renovation efforts": [[49, "3.2.-Mathematical-representation-of-renovation-efforts"]], "3.2.1. Hypotheses": [[49, "3.2.1.-Hypotheses"]], "3.2.2. Model": [[49, "3.2.2.-Model"]], "3.3. Optimization problem": [[49, "3.3.-Optimization-problem"]], "4. Dataset processing": [[49, "4.-Dataset-processing"]], "5. Solving the problem at hand": [[49, "5.-Solving-the-problem-at-hand"]], "Align all and Compute for Graphs": [[50, "Align-all-and-Compute-for-Graphs"]], "A primer in space, metric and aligners": [[50, "A-primer-in-space,-metric-and-aligners"]], "Frechet Mean": [[50, "Frechet-Mean"]], "Principal Components": [[50, "Principal-Components"]], "Regression": [[50, "Regression"]], "Mammals Grooming": [[51, "Mammals-Grooming"]], "Motivation and Dataset": [[51, "Motivation-and-Dataset"], [51, "id1"]], "GGRegression Fifa 2018 Player Passing Networks": [[51, "GGRegression-Fifa-2018-Player-Passing-Networks"]], "Tutorial: Sub-Riemannian geometry on the Heisenberg group": [[52, "Tutorial:-Sub-Riemannian-geometry-on-the-Heisenberg-group"]], "The Heisenberg group": [[52, "The-Heisenberg-group"]], "Defining the Heisenberg sub-Riemannian structure via a frame-field": [[52, "Defining-the-Heisenberg-sub-Riemannian-structure-via-a-frame-field"]], "Sub-Riemannian structures as limits of Riemannian structures": [[52, "Sub-Riemannian-structures-as-limits-of-Riemannian-structures"]], "Roadmap": [[53, "roadmap"]], "Purpose of this document": [[53, "purpose-of-this-document"]], "Statement of purpose: Geomstats in 2022": [[53, "statement-of-purpose-geomstats-in-2022"]], "Detailed Goals": [[53, "detailed-goals"]], "Tutorials": [[54, "tutorials"]], "Practical methods": [[54, "practical-methods"]], "Real world applications": [[54, "real-world-applications"]]}, "indexentries": {"autodiffnotimplementederror": [[0, "geomstats.exceptions.AutodiffNotImplementedError"]], "shapeerror": [[0, "geomstats.errors.ShapeError"]], "adapt_result() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.adapt_result"]], "adapt_types() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.adapt_types"]], "broadcast_to_multibatch() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.broadcast_to_multibatch"]], "check_belongs() (in module geomstats.errors)": [[0, "geomstats.errors.check_belongs"]], "check_integer() (in module geomstats.errors)": [[0, "geomstats.errors.check_integer"]], "check_is_batch() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.check_is_batch"]], "check_parameter_accepted_values() (in module geomstats.errors)": [[0, "geomstats.errors.check_parameter_accepted_values"]], "check_point_shape() (in module geomstats.errors)": [[0, "geomstats.errors.check_point_shape"]], "check_positive() (in module geomstats.errors)": [[0, "geomstats.errors.check_positive"]], "euler_step() (in module geomstats.integrator)": [[0, "geomstats.integrator.euler_step"]], "flip_determinant() (in module geomstats.algebra_utils)": [[0, "geomstats.algebra_utils.flip_determinant"]], "from_vector_to_diagonal_matrix() (in module geomstats.algebra_utils)": [[0, "geomstats.algebra_utils.from_vector_to_diagonal_matrix"]], "geomstats": [[0, "module-geomstats"]], "geomstats.algebra_utils": [[0, "module-geomstats.algebra_utils"]], "geomstats.errors": [[0, "module-geomstats.errors"]], "geomstats.exceptions": [[0, "module-geomstats.exceptions"]], "geomstats.integrator": [[0, "module-geomstats.integrator"]], "geomstats.vectorization": [[0, "module-geomstats.vectorization"]], "get_batch_shape() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.get_batch_shape"]], "get_initial_shapes() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.get_initial_shapes"]], "get_n_points() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.get_n_points"]], "get_types() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.get_types"]], "integrate() (in module geomstats.integrator)": [[0, "geomstats.integrator.integrate"]], "is_scalar() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.is_scalar"]], "leapfrog_step() (in module geomstats.integrator)": [[0, "geomstats.integrator.leapfrog_step"]], "module": [[0, "module-geomstats"], [0, "module-geomstats.algebra_utils"], [0, "module-geomstats.errors"], [0, "module-geomstats.exceptions"], [0, "module-geomstats.integrator"], [0, "module-geomstats.vectorization"], [1, "module-geomstats.datasets"], [1, "module-geomstats.datasets.prepare_emg_data"], [1, "module-geomstats.datasets.prepare_graph_data"], [1, "module-geomstats.datasets.utils"], [2, "module-geomstats.distributions"], [2, "module-geomstats.distributions.lognormal"], [3, "module-geomstats.geometry"], [3, "module-geomstats.geometry.base"], [3, "module-geomstats.geometry.complex_manifold"], [3, "module-geomstats.geometry.complex_matrices"], [3, "module-geomstats.geometry.complex_poincare_disk"], [3, "module-geomstats.geometry.complex_riemannian_metric"], [3, "module-geomstats.geometry.connection"], [3, "module-geomstats.geometry.diffeo"], [3, "module-geomstats.geometry.discrete_curves"], [3, "module-geomstats.geometry.discrete_surfaces"], [3, "module-geomstats.geometry.euclidean"], [3, "module-geomstats.geometry.fiber_bundle"], [3, "module-geomstats.geometry.full_rank_correlation_matrices"], [3, "module-geomstats.geometry.full_rank_matrices"], [3, "module-geomstats.geometry.functions"], [3, "module-geomstats.geometry.general_linear"], [3, "module-geomstats.geometry.grassmannian"], [3, "module-geomstats.geometry.heisenberg"], [3, "module-geomstats.geometry.hermitian"], [3, "module-geomstats.geometry.hermitian_matrices"], [3, "module-geomstats.geometry.hpd_matrices"], [3, "module-geomstats.geometry.hyperbolic"], [3, "module-geomstats.geometry.hyperboloid"], [3, "module-geomstats.geometry.hypersphere"], [3, "module-geomstats.geometry.invariant_metric"], [3, "module-geomstats.geometry.landmarks"], [3, "module-geomstats.geometry.lie_algebra"], [3, "module-geomstats.geometry.lie_group"], [3, "module-geomstats.geometry.lower_triangular_matrices"], [3, "module-geomstats.geometry.manifold"], [3, "module-geomstats.geometry.matrices"], [3, "module-geomstats.geometry.minkowski"], [3, "module-geomstats.geometry.nfold_manifold"], [3, "module-geomstats.geometry.open_hemisphere"], [3, "module-geomstats.geometry.poincare_ball"], [3, "module-geomstats.geometry.poincare_half_space"], [3, "module-geomstats.geometry.poincare_polydisk"], [3, "module-geomstats.geometry.positive_lower_triangular_matrices"], [3, "module-geomstats.geometry.positive_reals"], [3, "module-geomstats.geometry.pre_shape"], [3, "module-geomstats.geometry.product_hpd_and_siegel_disks"], [3, "module-geomstats.geometry.product_manifold"], [3, "module-geomstats.geometry.product_positive_reals_and_poincare_disks"], [3, "module-geomstats.geometry.pullback_metric"], [3, "module-geomstats.geometry.quotient_metric"], [3, "module-geomstats.geometry.rank_k_psd_matrices"], [3, "module-geomstats.geometry.riemannian_metric"], [3, "module-geomstats.geometry.sasaki_metric"], [3, "module-geomstats.geometry.scalar_product_metric"], [3, "module-geomstats.geometry.siegel"], [3, "module-geomstats.geometry.skew_symmetric_matrices"], [3, "module-geomstats.geometry.spd_matrices"], [3, "module-geomstats.geometry.special_euclidean"], [3, "module-geomstats.geometry.special_orthogonal"], [3, "module-geomstats.geometry.stiefel"], [3, "module-geomstats.geometry.sub_riemannian_metric"], [3, "module-geomstats.geometry.symmetric_matrices"], [4, "module-geomstats.geometry.stratified"], [4, "module-geomstats.geometry.stratified.graph_space"], [4, "module-geomstats.geometry.stratified.point_set"], [4, "module-geomstats.geometry.stratified.spider"], [4, "module-geomstats.geometry.stratified.wald_space"], [5, "module-geomstats.information_geometry"], [5, "module-geomstats.information_geometry.base"], [5, "module-geomstats.information_geometry.beta"], [5, "module-geomstats.information_geometry.binomial"], [5, "module-geomstats.information_geometry.categorical"], [5, "module-geomstats.information_geometry.dirichlet"], [5, "module-geomstats.information_geometry.exponential"], [5, "module-geomstats.information_geometry.fisher_rao_metric"], [5, "module-geomstats.information_geometry.gamma"], [5, "module-geomstats.information_geometry.geometric"], [5, "module-geomstats.information_geometry.multinomial"], [5, "module-geomstats.information_geometry.normal"], [5, "module-geomstats.information_geometry.poisson"], [6, "module-geomstats.learning"], [6, "module-geomstats.learning.aac"], [6, "module-geomstats.learning.agglomerative_hierarchical_clustering"], [6, "module-geomstats.learning.expectation_maximization"], [6, "module-geomstats.learning.exponential_barycenter"], [6, "module-geomstats.learning.frechet_mean"], [6, "module-geomstats.learning.geodesic_regression"], [6, "module-geomstats.learning.geometric_median"], [6, "module-geomstats.learning.incremental_frechet_mean"], [6, "module-geomstats.learning.kalman_filter"], [6, "module-geomstats.learning.kernel_density_estimation_classifier"], [6, "module-geomstats.learning.kmeans"], [6, "module-geomstats.learning.kmedoids"], [6, "module-geomstats.learning.knn"], [6, "module-geomstats.learning.mdm"], [6, "module-geomstats.learning.online_kmeans"], [6, "module-geomstats.learning.pca"], [6, "module-geomstats.learning.preprocessing"], [6, "module-geomstats.learning.radial_kernel_functions"], [6, "module-geomstats.learning.riemannian_mean_shift"], [6, "module-geomstats.learning.wrapped_gaussian_process"], [7, "module-geomstats.numerics"], [7, "module-geomstats.numerics.bvp"], [7, "module-geomstats.numerics.geodesic"], [7, "module-geomstats.numerics.ivp"], [7, "module-geomstats.numerics.optimizers"], [8, "module-geomstats.test"], [8, "module-geomstats.test.data"], [8, "module-geomstats.test.random"], [8, "module-geomstats.test.utils"], [8, "module-geomstats.test.vectorization"], [9, "module-geomstats.test_cases"], [11, "module-geomstats.test_cases.datasets"], [12, "module-geomstats.test_cases.distributions"], [13, "module-geomstats.test_cases.geometry"], [14, "module-geomstats.test_cases.geometry.stratified"], [15, "module-geomstats.test_cases.information_geometry"], [16, "module-geomstats.test_cases.learning"], [16, "module-geomstats.test_cases.learning.exponential_barycenter"], [17, "module-geomstats.test_cases.numerics"], [18, "module-geomstats.visualization"], [18, "module-geomstats.visualization.hyperbolic"], [18, "module-geomstats.visualization.hypersphere"], [18, "module-geomstats.visualization.poincare_polydisk"], [18, "module-geomstats.visualization.pre_shape"], [18, "module-geomstats.visualization.spd_matrices"], [18, "module-geomstats.visualization.special_euclidean"], [18, "module-geomstats.visualization.special_orthogonal"]], "repeat_out() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.repeat_out"]], "repeat_point() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.repeat_point"]], "rk2_step() (in module geomstats.integrator)": [[0, "geomstats.integrator.rk2_step"]], "rk4_step() (in module geomstats.integrator)": [[0, "geomstats.integrator.rk4_step"]], "rotate_points() (in module geomstats.algebra_utils)": [[0, "geomstats.algebra_utils.rotate_points"]], "squeeze_output_dim_0() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.squeeze_output_dim_0"]], "squeeze_output_dim_1() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.squeeze_output_dim_1"]], "symplectic_euler_step() (in module geomstats.integrator)": [[0, "geomstats.integrator.symplectic_euler_step"]], "taylor_exp_even_func() (in module geomstats.algebra_utils)": [[0, "geomstats.algebra_utils.taylor_exp_even_func"]], "vectorize_args() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.vectorize_args"]], "vectorize_kwargs() (in module geomstats.vectorization)": [[0, "geomstats.vectorization.vectorize_kwargs"]], "graph (class in geomstats.datasets.prepare_graph_data)": [[1, "geomstats.datasets.prepare_graph_data.Graph"]], "hyperbolicembedding (class in geomstats.datasets.prepare_graph_data)": [[1, "geomstats.datasets.prepare_graph_data.HyperbolicEmbedding"]], "timeseriescovariance (class in geomstats.datasets.prepare_emg_data)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance"]], "batches (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.batches"]], "covec (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.covec"]], "covs (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.covs"]], "data_dict (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.data_dict"]], "diags (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.diags"]], "edges (geomstats.datasets.prepare_graph_data.graph attribute)": [[1, "geomstats.datasets.prepare_graph_data.Graph.edges"]], "embed() (geomstats.datasets.prepare_graph_data.hyperbolicembedding method)": [[1, "geomstats.datasets.prepare_graph_data.HyperbolicEmbedding.embed"]], "geomstats.datasets": [[1, "module-geomstats.datasets"]], "geomstats.datasets.prepare_emg_data": [[1, "module-geomstats.datasets.prepare_emg_data"]], "geomstats.datasets.prepare_graph_data": [[1, "module-geomstats.datasets.prepare_graph_data"]], "geomstats.datasets.utils": [[1, "module-geomstats.datasets.utils"]], "grad_log_sigmoid() (geomstats.datasets.prepare_graph_data.hyperbolicembedding static method)": [[1, "geomstats.datasets.prepare_graph_data.HyperbolicEmbedding.grad_log_sigmoid"]], "grad_squared_distance() (geomstats.datasets.prepare_graph_data.hyperbolicembedding method)": [[1, "geomstats.datasets.prepare_graph_data.HyperbolicEmbedding.grad_squared_distance"]], "label_map (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.label_map"]], "labels (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.labels"]], "labels (geomstats.datasets.prepare_graph_data.graph attribute)": [[1, "geomstats.datasets.prepare_graph_data.Graph.labels"]], "load_cells() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_cells"]], "load_cities() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_cities"]], "load_connectomes() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_connectomes"]], "load_cube() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_cube"]], "load_emg() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_emg"]], "load_football() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_football"]], "load_hands() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_hands"]], "load_karate_graph() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_karate_graph"]], "load_leaves() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_leaves"]], "load_mammals() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_mammals"]], "load_optical_nerves() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_optical_nerves"]], "load_poses() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_poses"]], "load_random_graph() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_random_graph"]], "load_sao_paulo() (in module geomstats.datasets.utils)": [[1, "geomstats.datasets.utils.load_sao_paulo"]], "log_sigmoid() (geomstats.datasets.prepare_graph_data.hyperbolicembedding static method)": [[1, "geomstats.datasets.prepare_graph_data.HyperbolicEmbedding.log_sigmoid"]], "loss() (geomstats.datasets.prepare_graph_data.hyperbolicembedding method)": [[1, "geomstats.datasets.prepare_graph_data.HyperbolicEmbedding.loss"]], "margin (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.margin"]], "n_nodes (geomstats.datasets.prepare_graph_data.graph attribute)": [[1, "geomstats.datasets.prepare_graph_data.Graph.n_nodes"]], "n_steps (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.n_steps"]], "n_timeseries (geomstats.datasets.prepare_emg_data.timeseriescovariance attribute)": [[1, "geomstats.datasets.prepare_emg_data.TimeSeriesCovariance.n_timeseries"]], "random_walk() (geomstats.datasets.prepare_graph_data.graph method)": [[1, 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"geomstats.distributions.lognormal.LogNormalSPD.sample"]], "samples_sym() (geomstats.distributions.lognormal.lognormalspd method)": [[2, "geomstats.distributions.lognormal.LogNormalSPD.samples_sym"]], "autodiffdiffeo (class in geomstats.geometry.diffeo)": [[3, "geomstats.geometry.diffeo.AutodiffDiffeo"]], "biinvariantmetric (class in geomstats.geometry.invariant_metric)": [[3, "geomstats.geometry.invariant_metric.BiInvariantMetric"]], "bureswassersteinbundle (class in geomstats.geometry.rank_k_psd_matrices)": [[3, "geomstats.geometry.rank_k_psd_matrices.BuresWassersteinBundle"]], "choleskymap (class in geomstats.geometry.spd_matrices)": [[3, "geomstats.geometry.spd_matrices.CholeskyMap"]], "choleskymetric (class in geomstats.geometry.positive_lower_triangular_matrices)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.CholeskyMetric"]], "complexmanifold (class in geomstats.geometry.complex_manifold)": [[3, "geomstats.geometry.complex_manifold.ComplexManifold"]], 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(geomstats.geometry.pre_shape.preshapemetric method)": [[3, "geomstats.geometry.pre_shape.PreShapeMetric.exp"]], "exp() (geomstats.geometry.product_manifold.productriemannianmetric method)": [[3, "geomstats.geometry.product_manifold.ProductRiemannianMetric.exp"]], "exp() (geomstats.geometry.pullback_metric.pullbackdiffeometric method)": [[3, "geomstats.geometry.pullback_metric.PullbackDiffeoMetric.exp"]], "exp() (geomstats.geometry.quotient_metric.quotientmetric method)": [[3, "geomstats.geometry.quotient_metric.QuotientMetric.exp"]], "exp() (geomstats.geometry.sasaki_metric.sasakimetric method)": [[3, "geomstats.geometry.sasaki_metric.SasakiMetric.exp"]], "exp() (geomstats.geometry.siegel.siegelmetric method)": [[3, "geomstats.geometry.siegel.SiegelMetric.exp"]], "exp() (geomstats.geometry.spd_matrices.spdaffinemetric method)": [[3, "geomstats.geometry.spd_matrices.SPDAffineMetric.exp"]], "exp() (geomstats.geometry.spd_matrices.spdbureswassersteinmetric method)": [[3, "geomstats.geometry.spd_matrices.SPDBuresWassersteinMetric.exp"]], "exp() (geomstats.geometry.special_euclidean.specialeuclideanmatricescanonicalleftmetric method)": [[3, "geomstats.geometry.special_euclidean.SpecialEuclideanMatricesCanonicalLeftMetric.exp"]], "exp() (geomstats.geometry.stiefel.stiefelcanonicalmetric method)": [[3, "geomstats.geometry.stiefel.StiefelCanonicalMetric.exp"]], "exp() (geomstats.geometry.sub_riemannian_metric.subriemannianmetric method)": [[3, "geomstats.geometry.sub_riemannian_metric.SubRiemannianMetric.exp"]], "exp_at_zero() (geomstats.geometry.siegel.siegelmetric static method)": [[3, "geomstats.geometry.siegel.SiegelMetric.exp_at_zero"]], "exp_domain() (geomstats.geometry.hpd_matrices.hpdeuclideanmetric static method)": [[3, "geomstats.geometry.hpd_matrices.HPDEuclideanMetric.exp_domain"]], "exp_domain() (geomstats.geometry.spd_matrices.spdeuclideanmetric static method)": [[3, "geomstats.geometry.spd_matrices.SPDEuclideanMetric.exp_domain"]], "exp_from_identity() (geomstats.geometry.heisenberg.heisenbergvectors method)": [[3, "geomstats.geometry.heisenberg.HeisenbergVectors.exp_from_identity"]], "exp_from_identity() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.exp_from_identity"]], "exp_not_from_identity() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.exp_not_from_identity"]], "expmh() (in module geomstats.geometry.hermitian_matrices)": [[3, "geomstats.geometry.hermitian_matrices.expmh"]], "extrinsic_to_intrinsic_coords() (geomstats.geometry.base.levelset method)": [[3, "geomstats.geometry.base.LevelSet.extrinsic_to_intrinsic_coords"]], "extrinsic_to_intrinsic_coords() (geomstats.geometry.hyperboloid.hyperboloid method)": [[3, "geomstats.geometry.hyperboloid.Hyperboloid.extrinsic_to_intrinsic_coords"]], "face_areas() (geomstats.geometry.discrete_surfaces.discretesurfaces method)": [[3, "geomstats.geometry.discrete_surfaces.DiscreteSurfaces.face_areas"]], "flatten() (geomstats.geometry.matrices.matrices static method)": [[3, "geomstats.geometry.matrices.Matrices.flatten"]], "forward_difference() (in module geomstats.geometry.discrete_curves)": [[3, "geomstats.geometry.discrete_curves.forward_difference"]], "frobenius_product() (geomstats.geometry.complex_matrices.complexmatrices static method)": [[3, "geomstats.geometry.complex_matrices.ComplexMatrices.frobenius_product"]], "frobenius_product() (geomstats.geometry.matrices.matrices static method)": [[3, "geomstats.geometry.matrices.Matrices.frobenius_product"]], "from_covariance() (geomstats.geometry.full_rank_correlation_matrices.fullrankcorrelationmatrices class method)": [[3, "geomstats.geometry.full_rank_correlation_matrices.FullRankCorrelationMatrices.from_covariance"]], "geodesic() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.geodesic"]], "geodesic() (geomstats.geometry.product_manifold.productriemannianmetric method)": [[3, "geomstats.geometry.product_manifold.ProductRiemannianMetric.geodesic"]], "geodesic() (geomstats.geometry.pullback_metric.pullbackdiffeometric method)": [[3, "geomstats.geometry.pullback_metric.PullbackDiffeoMetric.geodesic"]], "geodesic() (geomstats.geometry.quotient_metric.quotientmetric method)": [[3, "geomstats.geometry.quotient_metric.QuotientMetric.geodesic"]], "geodesic() (geomstats.geometry.special_euclidean.specialeuclideanmatricescanonicalleftmetric method)": [[3, "geomstats.geometry.special_euclidean.SpecialEuclideanMatricesCanonicalLeftMetric.geodesic"]], "geodesic() (geomstats.geometry.sub_riemannian_metric.subriemannianmetric method)": [[3, "geomstats.geometry.sub_riemannian_metric.SubRiemannianMetric.geodesic"]], "geodesic_discrete() (geomstats.geometry.sasaki_metric.sasakimetric method)": [[3, "geomstats.geometry.sasaki_metric.SasakiMetric.geodesic_discrete"]], "geodesic_equation() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.geodesic_equation"]], "geomstats.geometry": [[3, "module-geomstats.geometry"]], "geomstats.geometry.base": [[3, "module-geomstats.geometry.base"]], "geomstats.geometry.complex_manifold": [[3, "module-geomstats.geometry.complex_manifold"]], "geomstats.geometry.complex_matrices": [[3, "module-geomstats.geometry.complex_matrices"]], "geomstats.geometry.complex_poincare_disk": [[3, "module-geomstats.geometry.complex_poincare_disk"]], "geomstats.geometry.complex_riemannian_metric": [[3, "module-geomstats.geometry.complex_riemannian_metric"]], "geomstats.geometry.connection": [[3, "module-geomstats.geometry.connection"]], "geomstats.geometry.diffeo": [[3, "module-geomstats.geometry.diffeo"]], "geomstats.geometry.discrete_curves": [[3, "module-geomstats.geometry.discrete_curves"]], "geomstats.geometry.discrete_surfaces": [[3, "module-geomstats.geometry.discrete_surfaces"]], "geomstats.geometry.euclidean": [[3, "module-geomstats.geometry.euclidean"]], "geomstats.geometry.fiber_bundle": [[3, "module-geomstats.geometry.fiber_bundle"]], "geomstats.geometry.full_rank_correlation_matrices": [[3, "module-geomstats.geometry.full_rank_correlation_matrices"]], "geomstats.geometry.full_rank_matrices": [[3, "module-geomstats.geometry.full_rank_matrices"]], "geomstats.geometry.functions": [[3, "module-geomstats.geometry.functions"]], "geomstats.geometry.general_linear": [[3, "module-geomstats.geometry.general_linear"]], "geomstats.geometry.grassmannian": [[3, "module-geomstats.geometry.grassmannian"]], "geomstats.geometry.heisenberg": [[3, "module-geomstats.geometry.heisenberg"]], "geomstats.geometry.hermitian": [[3, "module-geomstats.geometry.hermitian"]], "geomstats.geometry.hermitian_matrices": [[3, "module-geomstats.geometry.hermitian_matrices"]], "geomstats.geometry.hpd_matrices": [[3, "module-geomstats.geometry.hpd_matrices"]], "geomstats.geometry.hyperbolic": [[3, "module-geomstats.geometry.hyperbolic"]], "geomstats.geometry.hyperboloid": [[3, "module-geomstats.geometry.hyperboloid"]], "geomstats.geometry.hypersphere": [[3, "module-geomstats.geometry.hypersphere"]], "geomstats.geometry.invariant_metric": [[3, "module-geomstats.geometry.invariant_metric"]], "geomstats.geometry.landmarks": [[3, "module-geomstats.geometry.landmarks"]], "geomstats.geometry.lie_algebra": [[3, "module-geomstats.geometry.lie_algebra"]], "geomstats.geometry.lie_group": [[3, "module-geomstats.geometry.lie_group"]], "geomstats.geometry.lower_triangular_matrices": [[3, "module-geomstats.geometry.lower_triangular_matrices"]], "geomstats.geometry.manifold": [[3, "module-geomstats.geometry.manifold"]], "geomstats.geometry.matrices": [[3, "module-geomstats.geometry.matrices"]], "geomstats.geometry.minkowski": [[3, "module-geomstats.geometry.minkowski"]], "geomstats.geometry.nfold_manifold": [[3, "module-geomstats.geometry.nfold_manifold"]], "geomstats.geometry.open_hemisphere": [[3, "module-geomstats.geometry.open_hemisphere"]], "geomstats.geometry.poincare_ball": [[3, "module-geomstats.geometry.poincare_ball"]], "geomstats.geometry.poincare_half_space": [[3, "module-geomstats.geometry.poincare_half_space"]], "geomstats.geometry.poincare_polydisk": [[3, "module-geomstats.geometry.poincare_polydisk"]], "geomstats.geometry.positive_lower_triangular_matrices": [[3, "module-geomstats.geometry.positive_lower_triangular_matrices"]], "geomstats.geometry.positive_reals": [[3, "module-geomstats.geometry.positive_reals"]], "geomstats.geometry.pre_shape": [[3, "module-geomstats.geometry.pre_shape"]], "geomstats.geometry.product_hpd_and_siegel_disks": [[3, "module-geomstats.geometry.product_hpd_and_siegel_disks"]], "geomstats.geometry.product_manifold": [[3, "module-geomstats.geometry.product_manifold"]], "geomstats.geometry.product_positive_reals_and_poincare_disks": [[3, "module-geomstats.geometry.product_positive_reals_and_poincare_disks"]], "geomstats.geometry.pullback_metric": [[3, "module-geomstats.geometry.pullback_metric"]], "geomstats.geometry.quotient_metric": [[3, "module-geomstats.geometry.quotient_metric"]], "geomstats.geometry.rank_k_psd_matrices": [[3, "module-geomstats.geometry.rank_k_psd_matrices"]], "geomstats.geometry.riemannian_metric": [[3, "module-geomstats.geometry.riemannian_metric"]], "geomstats.geometry.sasaki_metric": [[3, "module-geomstats.geometry.sasaki_metric"]], "geomstats.geometry.scalar_product_metric": [[3, "module-geomstats.geometry.scalar_product_metric"]], "geomstats.geometry.siegel": [[3, "module-geomstats.geometry.siegel"]], "geomstats.geometry.skew_symmetric_matrices": [[3, "module-geomstats.geometry.skew_symmetric_matrices"]], "geomstats.geometry.spd_matrices": [[3, "module-geomstats.geometry.spd_matrices"]], "geomstats.geometry.special_euclidean": [[3, "module-geomstats.geometry.special_euclidean"]], "geomstats.geometry.special_orthogonal": [[3, "module-geomstats.geometry.special_orthogonal"]], "geomstats.geometry.stiefel": [[3, "module-geomstats.geometry.stiefel"]], "geomstats.geometry.sub_riemannian_metric": [[3, "module-geomstats.geometry.sub_riemannian_metric"]], "geomstats.geometry.symmetric_matrices": [[3, "module-geomstats.geometry.symmetric_matrices"]], "hamiltonian() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.hamiltonian"]], "hamiltonian() (geomstats.geometry.sub_riemannian_metric.subriemannianmetric method)": [[3, "geomstats.geometry.sub_riemannian_metric.SubRiemannianMetric.hamiltonian"]], "hessian_immersion() (geomstats.geometry.base.immersedset method)": [[3, "geomstats.geometry.base.ImmersedSet.hessian_immersion"]], "homogeneous_representation() (in module geomstats.geometry.special_euclidean)": [[3, "geomstats.geometry.special_euclidean.homogeneous_representation"]], "horizontal_lift() (geomstats.geometry.fiber_bundle.fiberbundle method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.horizontal_lift"]], "horizontal_lift() (geomstats.geometry.full_rank_correlation_matrices.correlationmatricesbundle method)": [[3, "geomstats.geometry.full_rank_correlation_matrices.CorrelationMatricesBundle.horizontal_lift"]], "horizontal_lift() (geomstats.geometry.rank_k_psd_matrices.bureswassersteinbundle method)": [[3, "geomstats.geometry.rank_k_psd_matrices.BuresWassersteinBundle.horizontal_lift"]], "horizontal_projection() (geomstats.geometry.discrete_curves.srvreparametrizationbundle method)": [[3, "geomstats.geometry.discrete_curves.SRVReparametrizationBundle.horizontal_projection"]], "horizontal_projection() (geomstats.geometry.discrete_curves.srvrotationbundle method)": [[3, "geomstats.geometry.discrete_curves.SRVRotationBundle.horizontal_projection"]], "horizontal_projection() (geomstats.geometry.discrete_curves.srvrotationreparametrizationbundle method)": [[3, "geomstats.geometry.discrete_curves.SRVRotationReparametrizationBundle.horizontal_projection"]], "horizontal_projection() (geomstats.geometry.fiber_bundle.fiberbundle method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.horizontal_projection"]], "identity (geomstats.geometry.euclidean.euclidean property)": [[3, "geomstats.geometry.euclidean.Euclidean.identity"]], "identity (geomstats.geometry.heisenberg.heisenbergvectors property)": [[3, "geomstats.geometry.heisenberg.HeisenbergVectors.identity"]], "identity (geomstats.geometry.hermitian.hermitian property)": [[3, "geomstats.geometry.hermitian.Hermitian.identity"]], "identity (geomstats.geometry.lie_group.liegroup property)": [[3, "geomstats.geometry.lie_group.LieGroup.identity"]], "identity (geomstats.geometry.lie_group.matrixliegroup property)": [[3, "geomstats.geometry.lie_group.MatrixLieGroup.identity"]], "immersion() (geomstats.geometry.base.immersedset method)": [[3, "geomstats.geometry.base.ImmersedSet.immersion"]], "injectivity_radius() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.grassmannian.grassmanniancanonicalmetric method)": [[3, "geomstats.geometry.grassmannian.GrassmannianCanonicalMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.hpd_matrices.hpdaffinemetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDAffineMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.hpd_matrices.hpdbureswassersteinmetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDBuresWassersteinMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.hpd_matrices.hpdeuclideanmetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDEuclideanMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.hyperboloid.hyperboloidmetric method)": [[3, "geomstats.geometry.hyperboloid.HyperboloidMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, "geomstats.geometry.hypersphere.HypersphereMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.invariant_metric.biinvariantmetric method)": [[3, "geomstats.geometry.invariant_metric.BiInvariantMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.minkowski.minkowskimetric method)": [[3, "geomstats.geometry.minkowski.MinkowskiMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.poincare_half_space.poincarehalfspacemetric method)": [[3, "geomstats.geometry.poincare_half_space.PoincareHalfSpaceMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.pre_shape.preshapemetric method)": [[3, "geomstats.geometry.pre_shape.PreShapeMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.spd_matrices.spdaffinemetric method)": [[3, "geomstats.geometry.spd_matrices.SPDAffineMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.spd_matrices.spdbureswassersteinmetric method)": [[3, "geomstats.geometry.spd_matrices.SPDBuresWassersteinMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.spd_matrices.spdeuclideanmetric method)": [[3, "geomstats.geometry.spd_matrices.SPDEuclideanMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.special_euclidean.specialeuclideanmatricescanonicalleftmetric method)": [[3, "geomstats.geometry.special_euclidean.SpecialEuclideanMatricesCanonicalLeftMetric.injectivity_radius"]], "injectivity_radius() (geomstats.geometry.stiefel.stiefelcanonicalmetric method)": [[3, "geomstats.geometry.stiefel.StiefelCanonicalMetric.injectivity_radius"]], "inner_coproduct() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.inner_coproduct"]], "inner_coproduct() (geomstats.geometry.sub_riemannian_metric.subriemannianmetric method)": [[3, "geomstats.geometry.sub_riemannian_metric.SubRiemannianMetric.inner_coproduct"]], "inner_product() (geomstats.geometry.complex_matrices.complexmatricesmetric static method)": [[3, "geomstats.geometry.complex_matrices.ComplexMatricesMetric.inner_product"]], "inner_product() (geomstats.geometry.complex_riemannian_metric.complexriemannianmetric method)": [[3, "geomstats.geometry.complex_riemannian_metric.ComplexRiemannianMetric.inner_product"]], "inner_product() (geomstats.geometry.discrete_curves.l2curvesmetric method)": [[3, "geomstats.geometry.discrete_curves.L2CurvesMetric.inner_product"]], "inner_product() (geomstats.geometry.discrete_surfaces.elasticmetric method)": [[3, "geomstats.geometry.discrete_surfaces.ElasticMetric.inner_product"]], "inner_product() (geomstats.geometry.euclidean.euclideanmetric method)": [[3, "geomstats.geometry.euclidean.EuclideanMetric.inner_product"]], "inner_product() (geomstats.geometry.functions.hilbertspheremetric method)": [[3, "geomstats.geometry.functions.HilbertSphereMetric.inner_product"]], "inner_product() (geomstats.geometry.hermitian.hermitianmetric static method)": [[3, "geomstats.geometry.hermitian.HermitianMetric.inner_product"]], "inner_product() (geomstats.geometry.hpd_matrices.hpdaffinemetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDAffineMetric.inner_product"]], "inner_product() (geomstats.geometry.hpd_matrices.hpdbureswassersteinmetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDBuresWassersteinMetric.inner_product"]], "inner_product() (geomstats.geometry.hyperboloid.hyperboloidmetric method)": [[3, "geomstats.geometry.hyperboloid.HyperboloidMetric.inner_product"]], "inner_product() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, "geomstats.geometry.hypersphere.HypersphereMetric.inner_product"]], "inner_product() (geomstats.geometry.invariant_metric.biinvariantmetric method)": [[3, "geomstats.geometry.invariant_metric.BiInvariantMetric.inner_product"]], "inner_product() (geomstats.geometry.matrices.matricesmetric method)": [[3, "geomstats.geometry.matrices.MatricesMetric.inner_product"]], "inner_product() (geomstats.geometry.minkowski.minkowskimetric method)": [[3, "geomstats.geometry.minkowski.MinkowskiMetric.inner_product"]], "inner_product() (geomstats.geometry.nfold_manifold.nfoldmetric method)": [[3, "geomstats.geometry.nfold_manifold.NFoldMetric.inner_product"]], "inner_product() (geomstats.geometry.poincare_half_space.poincarehalfspacemetric method)": [[3, "geomstats.geometry.poincare_half_space.PoincareHalfSpaceMetric.inner_product"]], "inner_product() (geomstats.geometry.positive_lower_triangular_matrices.choleskymetric class method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.CholeskyMetric.inner_product"]], "inner_product() (geomstats.geometry.pre_shape.preshapemetric method)": [[3, "geomstats.geometry.pre_shape.PreShapeMetric.inner_product"]], "inner_product() (geomstats.geometry.product_manifold.productriemannianmetric method)": [[3, "geomstats.geometry.product_manifold.ProductRiemannianMetric.inner_product"]], "inner_product() (geomstats.geometry.pullback_metric.pullbackdiffeometric method)": [[3, "geomstats.geometry.pullback_metric.PullbackDiffeoMetric.inner_product"]], "inner_product() (geomstats.geometry.quotient_metric.quotientmetric method)": [[3, "geomstats.geometry.quotient_metric.QuotientMetric.inner_product"]], "inner_product() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.inner_product"]], "inner_product() (geomstats.geometry.sasaki_metric.sasakimetric method)": [[3, "geomstats.geometry.sasaki_metric.SasakiMetric.inner_product"]], "inner_product() (geomstats.geometry.siegel.siegelmetric method)": [[3, "geomstats.geometry.siegel.SiegelMetric.inner_product"]], "inner_product() (geomstats.geometry.spd_matrices.spdaffinemetric method)": [[3, "geomstats.geometry.spd_matrices.SPDAffineMetric.inner_product"]], "inner_product() (geomstats.geometry.spd_matrices.spdbureswassersteinmetric method)": [[3, "geomstats.geometry.spd_matrices.SPDBuresWassersteinMetric.inner_product"]], "inner_product() (geomstats.geometry.special_euclidean.specialeuclideanmatricescanonicalleftmetric method)": [[3, "geomstats.geometry.special_euclidean.SpecialEuclideanMatricesCanonicalLeftMetric.inner_product"]], "inner_product() (geomstats.geometry.stiefel.stiefelcanonicalmetric method)": [[3, "geomstats.geometry.stiefel.StiefelCanonicalMetric.inner_product"]], "inner_product_at_identity() (geomstats.geometry.invariant_metric.biinvariantmetric method)": [[3, "geomstats.geometry.invariant_metric.BiInvariantMetric.inner_product_at_identity"]], "inner_product_at_identity() (geomstats.geometry.positive_lower_triangular_matrices.invariantpositivelowertriangularmatricesmetric method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.InvariantPositiveLowerTriangularMatricesMetric.inner_product_at_identity"]], "inner_product_derivative_matrix() (geomstats.geometry.pullback_metric.pullbackmetric method)": [[3, "geomstats.geometry.pullback_metric.PullbackMetric.inner_product_derivative_matrix"]], "inner_product_derivative_matrix() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.inner_product_derivative_matrix"]], "insert_origin() (geomstats.geometry.discrete_curves.discretecurvesstartingatorigin method)": [[3, "geomstats.geometry.discrete_curves.DiscreteCurvesStartingAtOrigin.insert_origin"]], "insert_zeros() (in module geomstats.geometry.discrete_curves)": [[3, "geomstats.geometry.discrete_curves.insert_zeros"]], "integrability_tensor() (geomstats.geometry.fiber_bundle.fiberbundle method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.integrability_tensor"]], "integrability_tensor() (geomstats.geometry.pre_shape.preshapespacebundle method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpaceBundle.integrability_tensor"]], "integrability_tensor_derivative() (geomstats.geometry.fiber_bundle.fiberbundle method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.integrability_tensor_derivative"]], "integrability_tensor_derivative() (geomstats.geometry.pre_shape.preshapespacebundle method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpaceBundle.integrability_tensor_derivative"]], "integrability_tensor_derivative_parallel() (geomstats.geometry.pre_shape.preshapespacebundle method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpaceBundle.integrability_tensor_derivative_parallel"]], "interpolate() (geomstats.geometry.discrete_curves.discretecurvesstartingatorigin method)": [[3, "geomstats.geometry.discrete_curves.DiscreteCurvesStartingAtOrigin.interpolate"]], "intrinsic_to_extrinsic_coords() (geomstats.geometry.base.levelset method)": [[3, "geomstats.geometry.base.LevelSet.intrinsic_to_extrinsic_coords"]], "intrinsic_to_extrinsic_coords() (geomstats.geometry.hyperboloid.hyperboloid method)": [[3, "geomstats.geometry.hyperboloid.Hyperboloid.intrinsic_to_extrinsic_coords"]], "intrinsic_to_extrinsic_coords() (geomstats.geometry.poincare_polydisk.poincarepolydisk static method)": [[3, "geomstats.geometry.poincare_polydisk.PoincarePolydisk.intrinsic_to_extrinsic_coords"]], "inverse() (geomstats.geometry.heisenberg.heisenbergvectors method)": [[3, "geomstats.geometry.heisenberg.HeisenbergVectors.inverse"]], "inverse() (geomstats.geometry.lie_group.liegroup class method)": [[3, "geomstats.geometry.lie_group.LieGroup.inverse"]], "inverse() (geomstats.geometry.lie_group.matrixliegroup class method)": [[3, "geomstats.geometry.lie_group.MatrixLieGroup.inverse"]], "inverse_diffeomorphism() (geomstats.geometry.diffeo.composeddiffeo method)": [[3, "geomstats.geometry.diffeo.ComposedDiffeo.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.diffeo.diffeo method)": [[3, "geomstats.geometry.diffeo.Diffeo.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.diffeo.reverseddiffeo method)": [[3, "geomstats.geometry.diffeo.ReversedDiffeo.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.discrete_curves.ftransform method)": [[3, "geomstats.geometry.discrete_curves.FTransform.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.discrete_curves.srvtransform method)": [[3, "geomstats.geometry.discrete_curves.SRVTransform.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.open_hemisphere.openhemispheretohyperboloiddiffeo method)": [[3, "geomstats.geometry.open_hemisphere.OpenHemisphereToHyperboloidDiffeo.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.positive_lower_triangular_matrices.unitnormedrowspltdiffeo method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.UnitNormedRowsPLTDiffeo.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.spd_matrices.choleskymap static method)": [[3, "geomstats.geometry.spd_matrices.CholeskyMap.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.spd_matrices.matrixpower method)": [[3, "geomstats.geometry.spd_matrices.MatrixPower.inverse_diffeomorphism"]], "inverse_diffeomorphism() (geomstats.geometry.spd_matrices.symmatrixlog class method)": [[3, "geomstats.geometry.spd_matrices.SymMatrixLog.inverse_diffeomorphism"]], "inverse_jacobian_diffeomorphism() (geomstats.geometry.diffeo.autodiffdiffeo method)": [[3, "geomstats.geometry.diffeo.AutodiffDiffeo.inverse_jacobian_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.diffeo.autodiffdiffeo method)": [[3, "geomstats.geometry.diffeo.AutodiffDiffeo.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.diffeo.composeddiffeo method)": [[3, "geomstats.geometry.diffeo.ComposedDiffeo.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.diffeo.diffeo method)": [[3, "geomstats.geometry.diffeo.Diffeo.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.diffeo.reverseddiffeo method)": [[3, "geomstats.geometry.diffeo.ReversedDiffeo.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.discrete_curves.srvtransform method)": [[3, "geomstats.geometry.discrete_curves.SRVTransform.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.open_hemisphere.openhemispheretohyperboloiddiffeo method)": [[3, "geomstats.geometry.open_hemisphere.OpenHemisphereToHyperboloidDiffeo.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.positive_lower_triangular_matrices.unitnormedrowspltdiffeo method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.UnitNormedRowsPLTDiffeo.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.spd_matrices.choleskymap class method)": [[3, "geomstats.geometry.spd_matrices.CholeskyMap.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.spd_matrices.matrixpower method)": [[3, "geomstats.geometry.spd_matrices.MatrixPower.inverse_tangent_diffeomorphism"]], "inverse_tangent_diffeomorphism() (geomstats.geometry.spd_matrices.symmatrixlog class method)": [[3, "geomstats.geometry.spd_matrices.SymMatrixLog.inverse_tangent_diffeomorphism"]], "is_centered() (geomstats.geometry.pre_shape.preshapespace static method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpace.is_centered"]], "is_diagonal() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_diagonal"]], "is_hermitian() (geomstats.geometry.complex_matrices.complexmatrices class method)": [[3, "geomstats.geometry.complex_matrices.ComplexMatrices.is_hermitian"]], "is_horizontal() (geomstats.geometry.fiber_bundle.fiberbundle method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.is_horizontal"]], "is_horizontal() (geomstats.geometry.pre_shape.preshapespacebundle method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpaceBundle.is_horizontal"]], "is_hpd() (geomstats.geometry.complex_matrices.complexmatrices class method)": [[3, "geomstats.geometry.complex_matrices.ComplexMatrices.is_hpd"]], "is_lower_triangular() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_lower_triangular"]], "is_pd() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_pd"]], "is_skew_hermitian() (geomstats.geometry.complex_matrices.complexmatrices class method)": [[3, "geomstats.geometry.complex_matrices.ComplexMatrices.is_skew_hermitian"]], "is_skew_symmetric() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_skew_symmetric"]], "is_spd() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_spd"]], "is_square() (geomstats.geometry.matrices.matrices static method)": [[3, "geomstats.geometry.matrices.Matrices.is_square"]], "is_strictly_lower_triangular() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_strictly_lower_triangular"]], "is_strictly_upper_triangular() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_strictly_upper_triangular"]], "is_symmetric() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_symmetric"]], "is_tangent() (geomstats.geometry.base.complexvectorspace method)": [[3, "geomstats.geometry.base.ComplexVectorSpace.is_tangent"]], "is_tangent() (geomstats.geometry.base.complexvectorspaceopenset method)": [[3, "geomstats.geometry.base.ComplexVectorSpaceOpenSet.is_tangent"]], "is_tangent() (geomstats.geometry.base.immersedset method)": [[3, "geomstats.geometry.base.ImmersedSet.is_tangent"]], "is_tangent() (geomstats.geometry.base.levelset method)": [[3, "geomstats.geometry.base.LevelSet.is_tangent"]], "is_tangent() (geomstats.geometry.base.openset method)": [[3, "geomstats.geometry.base.OpenSet.is_tangent"]], "is_tangent() (geomstats.geometry.base.vectorspace method)": [[3, "geomstats.geometry.base.VectorSpace.is_tangent"]], "is_tangent() (geomstats.geometry.base.vectorspaceopenset method)": [[3, "geomstats.geometry.base.VectorSpaceOpenSet.is_tangent"]], "is_tangent() (geomstats.geometry.complex_manifold.complexmanifold method)": [[3, "geomstats.geometry.complex_manifold.ComplexManifold.is_tangent"]], "is_tangent() (geomstats.geometry.discrete_surfaces.discretesurfaces method)": [[3, "geomstats.geometry.discrete_surfaces.DiscreteSurfaces.is_tangent"]], "is_tangent() (geomstats.geometry.functions.hilbertsphere method)": [[3, "geomstats.geometry.functions.HilbertSphere.is_tangent"]], "is_tangent() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.is_tangent"]], "is_tangent() (geomstats.geometry.lie_group.matrixliegroup method)": [[3, "geomstats.geometry.lie_group.MatrixLieGroup.is_tangent"]], "is_tangent() (geomstats.geometry.manifold.manifold method)": [[3, "geomstats.geometry.manifold.Manifold.is_tangent"]], "is_tangent() (geomstats.geometry.nfold_manifold.nfoldmanifold method)": [[3, "geomstats.geometry.nfold_manifold.NFoldManifold.is_tangent"]], "is_tangent() (geomstats.geometry.positive_lower_triangular_matrices.unitnormedrowspltmatrices method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.UnitNormedRowsPLTMatrices.is_tangent"]], "is_tangent() (geomstats.geometry.product_manifold.productmanifold method)": [[3, "geomstats.geometry.product_manifold.ProductManifold.is_tangent"]], "is_tangent() (geomstats.geometry.rank_k_psd_matrices.rankkpsdmatrices method)": [[3, "geomstats.geometry.rank_k_psd_matrices.RankKPSDMatrices.is_tangent"]], "is_tangent() (geomstats.geometry.sasaki_metric.tangentbundle method)": [[3, "geomstats.geometry.sasaki_metric.TangentBundle.is_tangent"]], "is_upper_triangular() (geomstats.geometry.matrices.matrices class method)": [[3, "geomstats.geometry.matrices.Matrices.is_upper_triangular"]], "is_vertical() (geomstats.geometry.fiber_bundle.fiberbundle method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.is_vertical"]], "isometry() (geomstats.geometry.siegel.siegelmetric static method)": [[3, "geomstats.geometry.siegel.SiegelMetric.isometry"]], "iterate() (geomstats.geometry.sub_riemannian_metric.subriemannianmetric static method)": [[3, "geomstats.geometry.sub_riemannian_metric.SubRiemannianMetric.iterate"]], "iterated_integrability_tensor_derivative_parallel() (geomstats.geometry.pre_shape.preshapespacebundle method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpaceBundle.iterated_integrability_tensor_derivative_parallel"]], "jacobian_diffeomorphism() (geomstats.geometry.diffeo.autodiffdiffeo method)": [[3, "geomstats.geometry.diffeo.AutodiffDiffeo.jacobian_diffeomorphism"]], "jacobian_immersion() (geomstats.geometry.base.immersedset method)": [[3, "geomstats.geometry.base.ImmersedSet.jacobian_immersion"]], "jacobian_translation() (geomstats.geometry.heisenberg.heisenbergvectors method)": [[3, "geomstats.geometry.heisenberg.HeisenbergVectors.jacobian_translation"]], "jacobian_translation() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.jacobian_translation"]], "k_landmarks (geomstats.geometry.landmarks.landmarks property)": [[3, "geomstats.geometry.landmarks.Landmarks.k_landmarks"]], "k_sampling_points (geomstats.geometry.discrete_curves.discretecurvesstartingatorigin property)": [[3, "geomstats.geometry.discrete_curves.DiscreteCurvesStartingAtOrigin.k_sampling_points"]], "ladder_parallel_transport() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.ladder_parallel_transport"]], "laplacian() (geomstats.geometry.discrete_surfaces.discretesurfaces method)": [[3, "geomstats.geometry.discrete_surfaces.DiscreteSurfaces.laplacian"]], "length() (geomstats.geometry.discrete_curves.discretecurvesstartingatorigin method)": [[3, "geomstats.geometry.discrete_curves.DiscreteCurvesStartingAtOrigin.length"]], "lie_algebra (geomstats.geometry.lie_group.liegroup attribute)": [[3, "geomstats.geometry.lie_group.LieGroup.lie_algebra"]], "lie_bracket() (geomstats.geometry.heisenberg.heisenbergvectors method)": [[3, "geomstats.geometry.heisenberg.HeisenbergVectors.lie_bracket"]], "lie_bracket() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.lie_bracket"]], "lie_bracket() (geomstats.geometry.lie_group.matrixliegroup method)": [[3, "geomstats.geometry.lie_group.MatrixLieGroup.lie_bracket"]], "lift() (geomstats.geometry.fiber_bundle.fiberbundle static method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.lift"]], "lift() (geomstats.geometry.rank_k_psd_matrices.bureswassersteinbundle method)": [[3, "geomstats.geometry.rank_k_psd_matrices.BuresWassersteinBundle.lift"]], "lifting() (geomstats.geometry.stiefel.stiefelcanonicalmetric method)": [[3, "geomstats.geometry.stiefel.StiefelCanonicalMetric.lifting"]], "log() (geomstats.geometry.complex_poincare_disk.complexpoincarediskmetric method)": [[3, "geomstats.geometry.complex_poincare_disk.ComplexPoincareDiskMetric.log"]], "log() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.log"]], "log() (geomstats.geometry.discrete_surfaces.elasticmetric method)": [[3, "geomstats.geometry.discrete_surfaces.ElasticMetric.log"]], "log() (geomstats.geometry.euclidean.euclideanmetric method)": [[3, "geomstats.geometry.euclidean.EuclideanMetric.log"]], "log() (geomstats.geometry.functions.hilbertspheremetric method)": [[3, "geomstats.geometry.functions.HilbertSphereMetric.log"]], "log() (geomstats.geometry.grassmannian.grassmanniancanonicalmetric method)": [[3, "geomstats.geometry.grassmannian.GrassmannianCanonicalMetric.log"]], "log() (geomstats.geometry.hermitian.hermitianmetric static method)": [[3, "geomstats.geometry.hermitian.HermitianMetric.log"]], "log() (geomstats.geometry.hpd_matrices.hpdaffinemetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDAffineMetric.log"]], "log() (geomstats.geometry.hpd_matrices.hpdbureswassersteinmetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDBuresWassersteinMetric.log"]], "log() (geomstats.geometry.hyperboloid.hyperboloidmetric method)": [[3, "geomstats.geometry.hyperboloid.HyperboloidMetric.log"]], "log() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, "geomstats.geometry.hypersphere.HypersphereMetric.log"]], "log() (geomstats.geometry.invariant_metric.biinvariantmetric method)": [[3, "geomstats.geometry.invariant_metric.BiInvariantMetric.log"]], "log() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.log"]], "log() (geomstats.geometry.lie_group.matrixliegroup class method)": [[3, "geomstats.geometry.lie_group.MatrixLieGroup.log"]], "log() (geomstats.geometry.minkowski.minkowskimetric method)": [[3, "geomstats.geometry.minkowski.MinkowskiMetric.log"]], "log() (geomstats.geometry.nfold_manifold.nfoldmetric method)": [[3, "geomstats.geometry.nfold_manifold.NFoldMetric.log"]], "log() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.log"]], "log() (geomstats.geometry.poincare_half_space.poincarehalfspacemetric method)": [[3, "geomstats.geometry.poincare_half_space.PoincareHalfSpaceMetric.log"]], "log() (geomstats.geometry.positive_lower_triangular_matrices.choleskymetric method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.CholeskyMetric.log"]], "log() (geomstats.geometry.positive_reals.positiverealsmetric static method)": [[3, "geomstats.geometry.positive_reals.PositiveRealsMetric.log"]], "log() (geomstats.geometry.pre_shape.preshapemetric method)": [[3, "geomstats.geometry.pre_shape.PreShapeMetric.log"]], "log() (geomstats.geometry.product_manifold.productriemannianmetric method)": [[3, "geomstats.geometry.product_manifold.ProductRiemannianMetric.log"]], "log() (geomstats.geometry.pullback_metric.pullbackdiffeometric method)": [[3, "geomstats.geometry.pullback_metric.PullbackDiffeoMetric.log"]], "log() (geomstats.geometry.quotient_metric.quotientmetric method)": [[3, "geomstats.geometry.quotient_metric.QuotientMetric.log"]], "log() (geomstats.geometry.sasaki_metric.sasakimetric method)": [[3, "geomstats.geometry.sasaki_metric.SasakiMetric.log"]], "log() (geomstats.geometry.siegel.siegelmetric method)": [[3, "geomstats.geometry.siegel.SiegelMetric.log"]], "log() (geomstats.geometry.spd_matrices.spdaffinemetric method)": [[3, "geomstats.geometry.spd_matrices.SPDAffineMetric.log"]], "log() (geomstats.geometry.spd_matrices.spdbureswassersteinmetric method)": [[3, "geomstats.geometry.spd_matrices.SPDBuresWassersteinMetric.log"]], "log() (geomstats.geometry.special_euclidean.specialeuclideanmatricescanonicalleftmetric method)": [[3, "geomstats.geometry.special_euclidean.SpecialEuclideanMatricesCanonicalLeftMetric.log"]], "log() (geomstats.geometry.stiefel.stiefelcanonicalmetric method)": [[3, "geomstats.geometry.stiefel.StiefelCanonicalMetric.log"]], "log_at_zero() (geomstats.geometry.siegel.siegelmetric static method)": [[3, "geomstats.geometry.siegel.SiegelMetric.log_at_zero"]], "log_from_identity() (geomstats.geometry.heisenberg.heisenbergvectors method)": [[3, "geomstats.geometry.heisenberg.HeisenbergVectors.log_from_identity"]], "log_from_identity() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.log_from_identity"]], "log_not_from_identity() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.log_not_from_identity"]], "logmh() (in module geomstats.geometry.spd_matrices)": [[3, "geomstats.geometry.spd_matrices.logmh"]], "matrix_representation() (geomstats.geometry.base.matrixvectorspace method)": [[3, "geomstats.geometry.base.MatrixVectorSpace.matrix_representation"]], "matrix_representation() (geomstats.geometry.general_linear.squarematrices method)": [[3, "geomstats.geometry.general_linear.SquareMatrices.matrix_representation"]], "matrix_representation() (geomstats.geometry.symmetric_matrices.symmetricmatrices static method)": [[3, "geomstats.geometry.symmetric_matrices.SymmetricMatrices.matrix_representation"]], "mean_curvature_vector() (geomstats.geometry.pullback_metric.pullbackmetric method)": [[3, "geomstats.geometry.pullback_metric.PullbackMetric.mean_curvature_vector"]], "metric_matrix() (geomstats.geometry.complex_poincare_disk.complexpoincarediskmetric method)": [[3, "geomstats.geometry.complex_poincare_disk.ComplexPoincareDiskMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.euclidean.euclideanmetric method)": [[3, "geomstats.geometry.euclidean.EuclideanMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.hermitian.hermitianmetric method)": [[3, "geomstats.geometry.hermitian.HermitianMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.hyperboloid.hyperboloidmetric method)": [[3, "geomstats.geometry.hyperboloid.HyperboloidMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, "geomstats.geometry.hypersphere.HypersphereMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.minkowski.minkowskimetric method)": [[3, "geomstats.geometry.minkowski.MinkowskiMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.nfold_manifold.nfoldmetric method)": [[3, "geomstats.geometry.nfold_manifold.NFoldMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.positive_reals.positiverealsmetric method)": [[3, "geomstats.geometry.positive_reals.PositiveRealsMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.product_manifold.productriemannianmetric method)": [[3, "geomstats.geometry.product_manifold.ProductRiemannianMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.pullback_metric.pullbackdiffeometric method)": [[3, "geomstats.geometry.pullback_metric.PullbackDiffeoMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.pullback_metric.pullbackmetric method)": [[3, "geomstats.geometry.pullback_metric.PullbackMetric.metric_matrix"]], "metric_matrix() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.metric_matrix"]], "minimize() (geomstats.geometry.sasaki_metric.gradientdescent method)": [[3, "geomstats.geometry.sasaki_metric.GradientDescent.minimize"]], "mobius_add() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.mobius_add"]], "mul() (geomstats.geometry.matrices.matrices static method)": [[3, "geomstats.geometry.matrices.Matrices.mul"]], "new() (geomstats.geometry.discrete_curves.discretecurvesstartingatorigin method)": [[3, "geomstats.geometry.discrete_curves.DiscreteCurvesStartingAtOrigin.new"]], "new() (geomstats.geometry.pre_shape.preshapespace method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpace.new"]], "norm() (geomstats.geometry.complex_matrices.complexmatricesmetric method)": [[3, "geomstats.geometry.complex_matrices.ComplexMatricesMetric.norm"]], "norm() (geomstats.geometry.euclidean.euclideanmetric method)": [[3, "geomstats.geometry.euclidean.EuclideanMetric.norm"]], "norm() (geomstats.geometry.hermitian.hermitianmetric static method)": [[3, "geomstats.geometry.hermitian.HermitianMetric.norm"]], "norm() (geomstats.geometry.matrices.matricesmetric method)": [[3, "geomstats.geometry.matrices.MatricesMetric.norm"]], "norm() (geomstats.geometry.pullback_metric.pullbackdiffeometric method)": [[3, "geomstats.geometry.pullback_metric.PullbackDiffeoMetric.norm"]], "norm() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.norm"]], "norm_factor_gradient() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, "geomstats.geometry.hypersphere.HypersphereMetric.norm_factor_gradient"]], "norm_factor_gradient() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.norm_factor_gradient"]], "normal_basis() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.normal_basis"]], "normalization_factor() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, "geomstats.geometry.hypersphere.HypersphereMetric.normalization_factor"]], "normalization_factor() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.normalization_factor"]], "normalize() (geomstats.geometry.discrete_curves.discretecurvesstartingatorigin method)": [[3, "geomstats.geometry.discrete_curves.DiscreteCurvesStartingAtOrigin.normalize"]], "normalize() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.normalize"]], "normals() (geomstats.geometry.discrete_surfaces.discretesurfaces method)": [[3, "geomstats.geometry.discrete_surfaces.DiscreteSurfaces.normals"]], "orbit() (geomstats.geometry.general_linear.generallinear class method)": [[3, "geomstats.geometry.general_linear.GeneralLinear.orbit"]], "parallel_transport() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.parallel_transport"]], "parallel_transport() (geomstats.geometry.euclidean.euclideanmetric method)": [[3, "geomstats.geometry.euclidean.EuclideanMetric.parallel_transport"]], "parallel_transport() (geomstats.geometry.grassmannian.grassmanniancanonicalmetric method)": [[3, "geomstats.geometry.grassmannian.GrassmannianCanonicalMetric.parallel_transport"]], "parallel_transport() (geomstats.geometry.hpd_matrices.hpdaffinemetric method)": [[3, 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(geomstats.geometry.discrete_surfaces.elasticmetric method)": [[3, "geomstats.geometry.discrete_surfaces.ElasticMetric.path_energy_per_time"]], "point_ndim (geomstats.geometry.manifold.manifold attribute)": [[3, "geomstats.geometry.manifold.Manifold.point_ndim"]], "pointwise_inner_product() (geomstats.geometry.nfold_manifold.nfoldmetric method)": [[3, "geomstats.geometry.nfold_manifold.NFoldMetric.pointwise_inner_product"]], "pointwise_norm() (geomstats.geometry.nfold_manifold.nfoldmetric method)": [[3, "geomstats.geometry.nfold_manifold.NFoldMetric.pointwise_norm"]], "powermh() (in module geomstats.geometry.hermitian_matrices)": [[3, "geomstats.geometry.hermitian_matrices.powermh"]], "projection() (geomstats.geometry.base.complexvectorspace static method)": [[3, "geomstats.geometry.base.ComplexVectorSpace.projection"]], "projection() (geomstats.geometry.base.complexvectorspaceopenset method)": [[3, "geomstats.geometry.base.ComplexVectorSpaceOpenSet.projection"]], "projection() 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(geomstats.geometry.functions.hilbertsphere method)": [[3, "geomstats.geometry.functions.HilbertSphere.random_point"]], "random_point() (geomstats.geometry.general_linear.generallinear method)": [[3, "geomstats.geometry.general_linear.GeneralLinear.random_point"]], "random_point() (geomstats.geometry.grassmannian.grassmannian method)": [[3, "geomstats.geometry.grassmannian.Grassmannian.random_point"]], "random_point() (geomstats.geometry.hermitian_matrices.hermitianmatrices method)": [[3, "geomstats.geometry.hermitian_matrices.HermitianMatrices.random_point"]], "random_point() (geomstats.geometry.hpd_matrices.hpdmatrices method)": [[3, "geomstats.geometry.hpd_matrices.HPDMatrices.random_point"]], "random_point() (geomstats.geometry.manifold.manifold method)": [[3, "geomstats.geometry.manifold.Manifold.random_point"]], "random_point() (geomstats.geometry.matrices.matrices method)": [[3, "geomstats.geometry.matrices.Matrices.random_point"]], "random_point() (geomstats.geometry.nfold_manifold.nfoldmanifold method)": [[3, "geomstats.geometry.nfold_manifold.NFoldManifold.random_point"]], "random_point() (geomstats.geometry.positive_lower_triangular_matrices.positivelowertriangularmatrices method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.PositiveLowerTriangularMatrices.random_point"]], "random_point() (geomstats.geometry.positive_reals.positivereals static method)": [[3, "geomstats.geometry.positive_reals.PositiveReals.random_point"]], "random_point() (geomstats.geometry.pre_shape.preshapespace method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpace.random_point"]], "random_point() (geomstats.geometry.product_manifold.productmanifold method)": [[3, "geomstats.geometry.product_manifold.ProductManifold.random_point"]], "random_point() (geomstats.geometry.rank_k_psd_matrices.rankkpsdmatrices method)": [[3, "geomstats.geometry.rank_k_psd_matrices.RankKPSDMatrices.random_point"]], "random_point() (geomstats.geometry.sasaki_metric.tangentbundle method)": [[3, "geomstats.geometry.sasaki_metric.TangentBundle.random_point"]], "random_point() (geomstats.geometry.siegel.siegel method)": [[3, "geomstats.geometry.siegel.Siegel.random_point"]], "random_point() (geomstats.geometry.skew_symmetric_matrices.skewsymmetricmatrices method)": [[3, "geomstats.geometry.skew_symmetric_matrices.SkewSymmetricMatrices.random_point"]], "random_point() (geomstats.geometry.spd_matrices.spdmatrices method)": [[3, "geomstats.geometry.spd_matrices.SPDMatrices.random_point"]], "random_point() (geomstats.geometry.special_euclidean.specialeuclideanmatricesliealgebra method)": [[3, "geomstats.geometry.special_euclidean.SpecialEuclideanMatricesLieAlgebra.random_point"]], "random_point() (geomstats.geometry.stiefel.stiefel method)": [[3, "geomstats.geometry.stiefel.Stiefel.random_point"]], "random_tangent_vec() (geomstats.geometry.base.diffeomorphicmanifold method)": [[3, "geomstats.geometry.base.DiffeomorphicManifold.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.complex_manifold.complexmanifold method)": [[3, "geomstats.geometry.complex_manifold.ComplexManifold.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.hpd_matrices.hpdmatrices method)": [[3, "geomstats.geometry.hpd_matrices.HPDMatrices.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.manifold.manifold method)": [[3, "geomstats.geometry.manifold.Manifold.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.product_manifold.productmanifold method)": [[3, "geomstats.geometry.product_manifold.ProductManifold.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.sasaki_metric.tangentbundle method)": [[3, "geomstats.geometry.sasaki_metric.TangentBundle.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.siegel.siegel method)": [[3, "geomstats.geometry.siegel.Siegel.random_tangent_vec"]], "random_tangent_vec() (geomstats.geometry.spd_matrices.spdmatrices method)": [[3, "geomstats.geometry.spd_matrices.SPDMatrices.random_tangent_vec"]], "random_uniform() (geomstats.geometry.grassmannian.grassmannian method)": [[3, "geomstats.geometry.grassmannian.Grassmannian.random_uniform"]], "random_uniform() (geomstats.geometry.pre_shape.preshapespace method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpace.random_uniform"]], "random_uniform() (geomstats.geometry.stiefel.stiefel method)": [[3, "geomstats.geometry.stiefel.Stiefel.random_uniform"]], "random_unit_tangent_vec() (geomstats.geometry.complex_riemannian_metric.complexriemannianmetric method)": [[3, "geomstats.geometry.complex_riemannian_metric.ComplexRiemannianMetric.random_unit_tangent_vec"]], "random_unit_tangent_vec() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.random_unit_tangent_vec"]], "regularize() (geomstats.geometry.base.diffeomorphicmanifold method)": [[3, "geomstats.geometry.base.DiffeomorphicManifold.regularize"]], "regularize() (geomstats.geometry.complex_manifold.complexmanifold method)": [[3, "geomstats.geometry.complex_manifold.ComplexManifold.regularize"]], "regularize() (geomstats.geometry.hyperboloid.hyperboloid method)": [[3, "geomstats.geometry.hyperboloid.Hyperboloid.regularize"]], "regularize() (geomstats.geometry.manifold.manifold method)": [[3, "geomstats.geometry.manifold.Manifold.regularize"]], "regularize() (geomstats.geometry.product_manifold.productmanifold method)": [[3, "geomstats.geometry.product_manifold.ProductManifold.regularize"]], "reshape() (geomstats.geometry.matrices.matrices method)": [[3, "geomstats.geometry.matrices.Matrices.reshape"]], "retraction() (geomstats.geometry.poincare_ball.poincareballmetric method)": [[3, "geomstats.geometry.poincare_ball.PoincareBallMetric.retraction"]], "retraction() (geomstats.geometry.stiefel.stiefelcanonicalmetric static method)": [[3, "geomstats.geometry.stiefel.StiefelCanonicalMetric.retraction"]], "ricci_tensor() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.ricci_tensor"]], "riemann_sum() (geomstats.geometry.discrete_curves.l2curvesmetric static method)": [[3, "geomstats.geometry.discrete_curves.L2CurvesMetric.riemann_sum"]], "riemann_tensor() (geomstats.geometry.connection.connection method)": [[3, "geomstats.geometry.connection.Connection.riemann_tensor"]], "riemannian_submersion() (geomstats.geometry.fiber_bundle.fiberbundle static method)": [[3, "geomstats.geometry.fiber_bundle.FiberBundle.riemannian_submersion"]], "riemannian_submersion() (geomstats.geometry.full_rank_correlation_matrices.correlationmatricesbundle static method)": [[3, "geomstats.geometry.full_rank_correlation_matrices.CorrelationMatricesBundle.riemannian_submersion"]], "riemannian_submersion() (geomstats.geometry.rank_k_psd_matrices.bureswassersteinbundle static method)": [[3, "geomstats.geometry.rank_k_psd_matrices.BuresWassersteinBundle.riemannian_submersion"]], "scalar_curvature() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.scalar_curvature"]], "second_centered_difference() (in module geomstats.geometry.discrete_curves)": [[3, "geomstats.geometry.discrete_curves.second_centered_difference"]], "second_fundamental_form() (geomstats.geometry.pullback_metric.pullbackmetric method)": [[3, "geomstats.geometry.pullback_metric.PullbackMetric.second_fundamental_form"]], "sectional_curvature() (geomstats.geometry.riemannian_metric.riemannianmetric method)": [[3, "geomstats.geometry.riemannian_metric.RiemannianMetric.sectional_curvature"]], "sectional_curvature() (geomstats.geometry.siegel.siegelmetric method)": [[3, "geomstats.geometry.siegel.SiegelMetric.sectional_curvature"]], "sectional_curvature_at_zero() (geomstats.geometry.siegel.siegelmetric method)": [[3, "geomstats.geometry.siegel.SiegelMetric.sectional_curvature_at_zero"]], "shape (geomstats.geometry.product_manifold.productriemannianmetric property)": [[3, "geomstats.geometry.product_manifold.ProductRiemannianMetric.shape"]], "squared_dist() (geomstats.geometry.complex_poincare_disk.complexpoincarediskmetric method)": [[3, "geomstats.geometry.complex_poincare_disk.ComplexPoincareDiskMetric.squared_dist"]], "squared_dist() (geomstats.geometry.grassmannian.grassmanniancanonicalmetric method)": [[3, "geomstats.geometry.grassmannian.GrassmannianCanonicalMetric.squared_dist"]], "squared_dist() (geomstats.geometry.hpd_matrices.hpdbureswassersteinmetric method)": [[3, "geomstats.geometry.hpd_matrices.HPDBuresWassersteinMetric.squared_dist"]], "squared_dist() (geomstats.geometry.hyperboloid.hyperboloidmetric method)": [[3, "geomstats.geometry.hyperboloid.HyperboloidMetric.squared_dist"]], "squared_dist() (geomstats.geometry.hypersphere.hyperspheremetric method)": [[3, 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index 2d9985781..796dd0857 100644
--- a/tutorials/index.html
+++ b/tutorials/index.html
@@ -49,7 +49,7 @@
     
   
   
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+