Principal Stiefel Coordinates (PSC)

Python code for the paper "Equivariant Dimensionality Reduction on Stiefel Manifolds" by Andrew Lee, Harlin Lee, Jose Perea, Nikolas Schonsheck, and Madeleine Weinstein.

What do I do first?

For the easiest example,

git clone https://github.com/HarlinLee/PSC.git
cd PSC
python demo.py

You may be asked to install certain packages, such as

pip install pymanopt==2.2.0
pip install autograd==1.6.2
pip install geomstats==2.6.0

What's inside?

The PSC/ package contains:

1. projections.py: Our PSC algorithm including manopt gradient descent and $\pi_\alpha$.

2. utils.py: Helper code including calculation of projection error.

3. comparison.py: Helper functions for PGA comparison.

4. plots.py: Plotting functions for PGA comparison.

The experiments/ folder contains code to recreate experiments and figures produced in the paper. The Jupyter notebooks were tested in Google Colab, so if you appropriately edit the variable DRIVE_PATH and output_folder, every experiment should be reproducible in either Google Colab or your local machine.

1. low_dim.ipynb: Low-dimensional example with $k=1, n=2, N=3$ for PSC demonstration. Figures lowdim-opt.pdf, lowdim_piy_0.8.pdf, lowdim_y_0.8.pdf, lowdim_yhat_0.8.pdf are outputs of this code with noise level $\epsilon=0.8$.

2. variance_comparison.ipynb: Comparison with PGA. var_data.pkl is saved output. Check comparison.py and plots.py for more details.

3. brain.ipynb: Brain connectivity matrix experiment. Uses matlab data saved in the folder connectivity_matrices and saves the plot brain_projected.pdf.

4. neuron/

   • create_neural_response.ipynb: A Julia file that generates data according to the neuronal stimulus space model. It uses auxilary functions in julia_utilities.jl. This makes files such as neurons.h5, random_walk.h5 (or 100_neurons_13k_steps_nonuniform_half_random_walk.h5), and response_matrix.h5. centered_normalized_response_matrix_100_neurons_13k_steps_nonuniform_half_random_walk.h5 is the centered and $\ell_2$-normalized response matrix (see code at top of stimulus_space_model.ipynb).

   • stimulus_space_model.ipynb: Applies PSC, MDS, and persistent cohomology on the response matrix. Since MDS takes a while, MDS_script.py can be used as an alternative for running MDS inside the Jupyter notebook. Figures psc_angle.pdf, psc_path.pdf, comparisons-others.pdf, comparisons-path.pdf are generated, and dimensionality reduction results circ_coords.h5, mds.h5, PSC.pkl are saved.

5. video/

   • video-clustering.ipynb: Video clustering experiment.