Regression-Based Elastic Metric Learning

Official implementation of the paper “Regression-Based Elastic Metric Learning on Shape Spaces of Cell Curves”.

[Paper] published at [NeurIPS Workshop Learning Meaningful Representations of Life]

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⭐️ Overview of Goals

• Regression-Based Elastic Metric Learning is a machine learning tool designed to improve analysis of discrete parameterizations of 2D cures changing over time.
• Specifically, we optimize geodesic regression analysis by learning the elastic metric parameters that model a given data trajectory close to a geodesic.

Left: A trajectory may follow a geodesic as calculated by one metric but not follow a geodesic as calculated by another metric. Our paradigm learns the elastic metric (parameterized by $a$) that best models the data trajectory as a geodesic on the manifold of discrete curves. Right: true cell trajectory overlaid with 1) cells predicted by a regression which utilizes our paradigm’s learned metric parameter ($a*$) 2) cells predicted by a square-root-velocity (SRV) regression. Regression predictions using the SRV metric (red) do not match the data trajectory (blue), but our algorithm’s $a*$ predicts the data trajectory perfectly: our prediction (green) perfectly overlays the data trajectory (blue).

Elastic Metric

• We consider a family of elastic metrics given by:

$$g^{a, b}_c(h, k) = a^2 \int_{0}^1\langle D_sh, N\rangle\langle D_sk, N\rangle ds + b^2 \int_{0}^1\langle D_sh, T\rangle\langle D_sk, T\rangle ds$$

• We use the elastic metric implementation in Geomstats.
• The elastic metric is parameterized by $a$ and $b$ which quantify how much two shapes are "stretched" or "bent" compared to each other, respectively.
• Changing $a$ and $b$ of the elastic metric changes the distance between various points on the manifold of discrete curves: the space where we analyze curves. As such, changing $a$ and $b$ changes the nature of geodesics on the manifold of discrete curves.

• Note that the ratio $a/b$ is sufficient to describe variationos of $ g^{a, b}$. Thus, we set $b=0.5$, as varying $b$ only changes units of the calculation.
• Our paradigm learns the $a*$ (and therefore the ratio $a*/b$) which models the data trajectory as being closest to a geodesic, as evaluated by the coefficient of determination $R^2$.
• We use a gradient ascent algorithm, along with our derived analytical expression of $R^2$ in terms of $a$, to find the $a*$ which maximizes $R^2$ for a given trajectory.

Experiments

• We apply our paradigm to data trajectories of cell outlines changing over time
• For each experiement, we generate a semi-synthetic data trajectory by drawing a geodesic between two real cancer cells

• We create the trajectory with a predetermined 1) number of cells 2) number of sampling points (how many times each cell outline is sampled) 3) amount of noise 4) "true $a$" (the metric used to draw the geodesic between real cancer cells). Note that because the semi-synthetic geodesic is drawn with the metric parameter $a_{true}$, the metric parameter $a_{true}$ WILL model the trajectory as a geodesic.
• Thus, the gradient ascent learning scheme aims to learn an $a*$ close to $a_{true}$.

Results

• We compare the predictive power of $a*$ regression to the predictive power of regression with the square-root-velocity (SRV) metric, which is a special case of the elastic metric where $a = 1$ and $b = 0.5$.
• Performing geodesic regression with our learned $a*$ metric parameter improves predictive power to geodesic regression, as geodesic regression is more accurate when the data trajectory is close to a geodesic.

🌎 Bibtex

If this code is useful to your research, please cite:

@misc{https://doi.org/10.48550/arxiv.2210.01932,
  doi = {10.48550/ARXIV.2210.01932},

  url = {https://arxiv.org/abs/2210.01932},

  author = {Myers, Adele and Miolane, Nina},

  keywords = {Machine Learning (cs.LG), FOS: Computer and information sciences, FOS: Computer and information sciences},

  title = {Regression-Based Elastic Metric Learning on Shape Spaces of Elastic Curves},

  publisher = {arXiv},

  year = {2022},

  copyright = {Creative Commons Attribution 4.0 International}
}

🏡 Installation

This codes runs on Python 3.8. We recommend using Anaconda for easy installation. To create the necessary conda environment, run:

cd dyn
conda env create -f environment.yml
conda activate dyn

🏃‍♀️ Run the Code

We use Wandb to keep track of our runs. To launch a new run, follow the steps below.

1. Set up Wandb logging.

Wandb is a powerful tool for logging performance during training, as well as animation artifacts. To use it, simply create an account, then run:

wandb login

to sign into your account.

2. Create a new project in Wandb.

Create a new project in Wandb called "metric-learning-TEST".

3. Specify hyperparameters in default_config.py.

"main.py" of our program runs through every combination of hyperparameters specified in "default_config.py". Change any of the following hyperparameters:

• a_true: "a" is the metric parameter used to generate the synthetic shape trajectory. This is the metric parameter that the code is trying to learn.
• n_sampling_points: the number of points in each cell shape
• n_times: the number of cell shapes in the data trajectory
• noise_std: the amount of noise added to the synthetic data. (how "noisy" is each cell shape)
• percent_train: percent of the data trajectory used to train the regression model in our code
• percent_val: percent of the data trajectory used to validate the regression model (using the coefficient of determination $R^2$) and learn a* (our code's best estimate of a_true). Note: the rest of the data trajectory that is not used to train or validate the model will be used to test the predictive power of a* regression against the baseline square-root-velocity (SRV) metric regression.
• dataset: you can either test the code on a synthetic geodesic between a circle and an ellipse or a semi-synthetic geodesic between two real cancer cells.

4. Run!

For a single run, use the command:

python main.py

This will initiate runs with every combination of hyperparameters detailed in default_config.py.

5. 👀 See Results.

You can see all of your runs by logging into the Wandb webpage and looking under your project name "metric-learning-TEST". Our code automatically names each run as

<dataset-used>_<a_true>_<gradient-ascent-initiation>_mt1_<n_times>_<n_sampling_points>_<noise_std>_<time-of-run>

👩‍🔧 Authors

Adele Myers

Nina Miolane