From b798c3bb8008fea4b8a15c67f36fe8136ff6673b Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Thu, 21 Nov 2024 09:52:34 +0000 Subject: [PATCH] build based on dd25a16 --- previews/PR255/.documenter-siteinfo.json | 2 +- previews/PR255/tensors.html | 12 ++++++------ previews/PR255/transformations-29bd62b7.png | Bin 18030 -> 0 bytes previews/PR255/transformations-75427db6.png | Bin 0 -> 18024 bytes previews/PR255/transformations.html | 2 +- 5 files changed, 8 insertions(+), 8 deletions(-) delete mode 100644 previews/PR255/transformations-29bd62b7.png create mode 100644 previews/PR255/transformations-75427db6.png diff --git a/previews/PR255/.documenter-siteinfo.json b/previews/PR255/.documenter-siteinfo.json index ee8118fa..07921b83 100644 --- a/previews/PR255/.documenter-siteinfo.json +++ b/previews/PR255/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.11.1","generation_timestamp":"2024-11-21T09:52:23","documenter_version":"1.8.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.11.1","generation_timestamp":"2024-11-21T09:52:28","documenter_version":"1.8.0"}} \ No newline at end of file diff --git a/previews/PR255/tensors.html b/previews/PR255/tensors.html index 7555625a..04f5968d 100644 --- a/previews/PR255/tensors.html +++ b/previews/PR255/tensors.html @@ -1,11 +1,11 @@ Tensors · Tenet.jl
GitHub

Tensors

There are many jokes[1] about how to define a tensor. The definition we are giving here might not be the most correct one, but it is good enough for our use case (don't kill me please, mathematicians). A tensor $T$ of order[2] $n$ is a multilinear[3] application between $n$ vector spaces over a field $\mathcal{F}$.

\[T : \mathcal{F}^{\dim(1)} \times \dots \times \mathcal{F}^{\dim(n)} \mapsto \mathcal{F}\]

In layman's terms, it is a linear function whose inputs are vectors and the output is a scalar number.

\[T(\mathbf{v}^{(1)}, \dots, \mathbf{v}^{(n)}) = c \in \mathcal{F} \qquad\qquad \forall i, \mathbf{v}^{(i)} \in \mathcal{F}^{\dim(i)}\]

Tensor algebra is a higher-order generalization of linear algebra, where scalar numbers can be viewed as order-0 tensors, vectors as order-1 tensors, matrices as order-2 tensors, ...

Letters are used to identify each of the vector spaces the tensor relates to. In computer science, you would intuitively think of tensors as "n-dimensional arrays with named dimensions".

\[T_{ijk} \iff \mathtt{T[i,j,k]}\]

The Tensor type

In Tenet, a tensor is represented by the Tensor type, which wraps an array and a list of symbols. As it subtypes AbstractArray, many array operations can be dispatched to it.

You can create a Tensor by passing an array and a list of Symbols that name indices.

julia> Tᵢⱼₖ = Tensor(rand(3,5,2), (:i,:j,:k))3×5×2 Tensor{Float64, 3, Array{Float64, 3}}:
 [:, :, 1] =
- 0.948613  0.829285  0.819368   0.381128   0.145871
- 0.886361  0.161633  0.0426221  0.0273748  0.570567
- 0.458981  0.620724  0.555383   0.0668311  0.630917
+ 0.591392  0.499633  0.675227  0.186081   0.505033
+ 0.361674  0.264754  0.556073  0.745471   0.655058
+ 0.834783  0.989294  0.981009  0.0563247  0.449567
 
 [:, :, 2] =
- 0.431614  0.494151   0.298355  0.393384  0.276856
- 0.885448  0.0992291  0.581982  0.954357  0.384265
- 0.834856  0.104122   0.442789  0.345902  0.335081

The dimensionality or size of each index can be consulted using the size function.

Base.size — Method
Base.size(::Tensor[, i])

Return the size of the underlying array or the dimension i (specified by Symbol or Integer).

source
julia> size(Tᵢⱼₖ)(3, 5, 2)
julia> size(Tᵢⱼₖ, :j)5
julia> length(Tᵢⱼₖ)30

Operations

Contraction

Graphs.LinAlg.contract — Method
contract(a::Tensor[, b::Tensor]; dims=nonunique([inds(a)..., inds(b)...]))

Perform tensor contraction operation.

source

Factorizations

LinearAlgebra.svd — Method
LinearAlgebra.svd(tensor::Tensor; left_inds, right_inds, virtualind, kwargs...)

Perform SVD factorization on a tensor.

Keyword arguments

  • left_inds: left indices to be used in the SVD factorization. Defaults to all indices of t except right_inds.
  • right_inds: right indices to be used in the SVD factorization. Defaults to all indices of t except left_inds.
  • virtualind: name of the virtual bond. Defaults to a random Symbol.
source
LinearAlgebra.qr — Method
LinearAlgebra.qr(tensor::Tensor; left_inds, right_inds, virtualind, kwargs...)

Perform QR factorization on a tensor.

Keyword arguments

  • left_inds: left indices to be used in the QR factorization. Defaults to all indices of t except right_inds.
  • right_inds: right indices to be used in the QR factorization. Defaults to all indices of t except left_inds.
  • virtualind: name of the virtual bond. Defaults to a random Symbol.
source
LinearAlgebra.lu — Method
LinearAlgebra.lu(tensor::Tensor; left_inds, right_inds, virtualind, kwargs...)

Perform LU factorization on a tensor.

Keyword arguments

  • left_inds: left indices to be used in the LU factorization. Defaults to all indices of t except right_inds.
  • right_inds: right indices to be used in the LU factorization. Defaults to all indices of t except left_inds.
  • virtualind: name of the virtual bond. Defaults to a random Symbol.
source
  • 1For example, recursive definitions like a tensor is whatever that transforms as a tensor.
  • 2The order of a tensor may also be known as rank or dimensionality in other fields. However, these can be missleading, since it has nothing to do with the rank of linear algebra nor with the dimensionality of a vector space. We prefer to use word order.
  • 3Meaning that the relationships between the output and the inputs, and the inputs between them, are linear.
+ 0.365638 0.0552327 0.854676 0.958947 0.664245 + 0.904802 0.762236 0.534377 0.226843 0.752238 + 0.26818 0.989104 0.231947 0.167473 0.379442

The dimensionality or size of each index can be consulted using the size function.

Base.size — Method
Base.size(::Tensor[, i])

Return the size of the underlying array or the dimension i (specified by Symbol or Integer).

source
julia> size(Tᵢⱼₖ)(3, 5, 2)
julia> size(Tᵢⱼₖ, :j)5
julia> length(Tᵢⱼₖ)30

Operations

Contraction

Graphs.LinAlg.contract — Method
contract(a::Tensor[, b::Tensor]; dims=nonunique([inds(a)..., inds(b)...]))

Perform tensor contraction operation.

source

Factorizations

LinearAlgebra.svd — Method
LinearAlgebra.svd(tensor::Tensor; left_inds, right_inds, virtualind, kwargs...)

Perform SVD factorization on a tensor.

Keyword arguments

  • left_inds: left indices to be used in the SVD factorization. Defaults to all indices of t except right_inds.
  • right_inds: right indices to be used in the SVD factorization. Defaults to all indices of t except left_inds.
  • virtualind: name of the virtual bond. Defaults to a random Symbol.
source
LinearAlgebra.qr — Method
LinearAlgebra.qr(tensor::Tensor; left_inds, right_inds, virtualind, kwargs...)

Perform QR factorization on a tensor.

Keyword arguments

  • left_inds: left indices to be used in the QR factorization. Defaults to all indices of t except right_inds.
  • right_inds: right indices to be used in the QR factorization. Defaults to all indices of t except left_inds.
  • virtualind: name of the virtual bond. Defaults to a random Symbol.
source
LinearAlgebra.lu — Method
LinearAlgebra.lu(tensor::Tensor; left_inds, right_inds, virtualind, kwargs...)

Perform LU factorization on a tensor.

Keyword arguments

  • left_inds: left indices to be used in the LU factorization. Defaults to all indices of t except right_inds.
  • right_inds: right indices to be used in the LU factorization. Defaults to all indices of t except left_inds.
  • virtualind: name of the virtual bond. Defaults to a random Symbol.
source
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GitHub

Transformations

In tensor network computations, it is good practice to apply various transformations to simplify the network structure, reduce computational cost, or prepare the network for further operations. These transformations modify the network's structure locally by permuting, contracting, factoring or truncating tensors.

A crucial reason why these methods are indispensable lies in their ability to drastically reduce the problem size of the contraction path search and also the contraction. This doesn't necessarily involve reducing the maximum rank of the Tensor Network itself, but more importantly, it reduces the size (or rank) of the involved tensors.

Our approach is based in (Gray and Kourtis, 2021), which can also be found in quimb.

In Tenet, we provide a set of predefined transformations which you can apply to your TensorNetwork using both the transform/transform! functions.

Tenet.transform — Function
transform(tn::TensorNetwork, config::Transformation)
 transform(tn::TensorNetwork, configs)

Return a new TensorNetwork where some Transformation has been performed into it.

See also: transform!.

source
Tenet.transform! — Function
transform!(tn::TensorNetwork, config::Transformation)
-transform!(tn::TensorNetwork, configs)

In-place version of transform.

source

Available transformations

Hyperindex converter

Tenet.HyperFlatten — Type
HyperFlatten <: Transformation

Convert hyperindices to COPY-tensors, represented by DeltaArrays. This transformation is always used by default when visualizing a TensorNetwork with plot.

See also: HyperGroup.

source

Contraction simplification

Example block output

Diagonal reduction

Tenet.DiagonalReduction — Type
DiagonalReduction <: Transformation

Reduce the dimension of a Tensor in a TensorNetwork when it has a pair of indices that fulfil a diagonal structure.

Keyword Arguments

  • atol Absolute tolerance. Defaults to 1e-12.
source
Example block output

Anti-diagonal reduction

Tenet.AntiDiagonalGauging — Type
AntiDiagonalGauging <: Transformation

Reverse the order of tensor indices that fulfill the anti-diagonal condition. While this transformation doesn't directly enhance computational efficiency, it sets up the TensorNetwork for other operations that do.

Keyword Arguments

  • atol Absolute tolerance. Defaults to 1e-12.
  • skip List of indices to skip. Defaults to [].
source

Dimension truncation

Tenet.Truncate — Type
Truncate <: Transformation

Truncate the dimension of a Tensor in a TensorNetwork when it contains columns with all elements smaller than atol.

Keyword Arguments

  • atol Absolute tolerance. Defaults to 1e-12.
  • skip List of indices to skip. Defaults to [].
source
Example block output

Split simplification

Tenet.SplitSimplification — Type
SplitSimplification <: Transformation

Reduce the rank of tensors in the TensorNetwork by decomposing them using the Singular Value Decomposition (SVD). Tensors whose factorization do not increase the maximum rank of the network are left decomposed.

Keyword Arguments

  • atol Absolute tolerance. Defaults to 1e-10.
source
Example block output
+transform!(tn::TensorNetwork, configs)

In-place version of transform.

source

Available transformations

Hyperindex converter

Tenet.HyperFlatten — Type
HyperFlatten <: Transformation

Convert hyperindices to COPY-tensors, represented by DeltaArrays. This transformation is always used by default when visualizing a TensorNetwork with plot.

See also: HyperGroup.

source

Contraction simplification

Example block output

Diagonal reduction

Tenet.DiagonalReduction — Type
DiagonalReduction <: Transformation

Reduce the dimension of a Tensor in a TensorNetwork when it has a pair of indices that fulfil a diagonal structure.

Keyword Arguments

  • atol Absolute tolerance. Defaults to 1e-12.
source
Example block output

Anti-diagonal reduction

Tenet.AntiDiagonalGauging — Type
AntiDiagonalGauging <: Transformation

Reverse the order of tensor indices that fulfill the anti-diagonal condition. While this transformation doesn't directly enhance computational efficiency, it sets up the TensorNetwork for other operations that do.

Keyword Arguments

  • atol Absolute tolerance. Defaults to 1e-12.
  • skip List of indices to skip. Defaults to [].
source

Dimension truncation

Tenet.Truncate — Type
Truncate <: Transformation

Truncate the dimension of a Tensor in a TensorNetwork when it contains columns with all elements smaller than atol.

Keyword Arguments

  • atol Absolute tolerance. Defaults to 1e-12.
  • skip List of indices to skip. Defaults to [].
source
Example block output

Split simplification

Tenet.SplitSimplification — Type
SplitSimplification <: Transformation

Reduce the rank of tensors in the TensorNetwork by decomposing them using the Singular Value Decomposition (SVD). Tensors whose factorization do not increase the maximum rank of the network are left decomposed.

Keyword Arguments

  • atol Absolute tolerance. Defaults to 1e-10.
source
Example block output