Code for reproducing the results from the paper Product Manifold Learning, in AISTATS 2021.

Before running this project, make sure to pip install the following packages:

• cvxpy
• scipy
• scikit-learn
• numpy
• matplotlib

This project was developed using Python 3.7.0.

To generate a dataset, create an appropriate .json file in the params folder. Then, run

python generate_data.py <path/to/params/file1> <path/to/params/file2> ...

Note that multiple datasets can be generated at once by passing a list of .json files. For example,

python generate_data.py params/params_rectangle3d.json params/params_torus.json

will generate two separate datasets. The datasets will be pickle files located in the data/ folder.

Specifications for the datasets used in the paper can be changed by changing the keys in the params file.

The params files for geometric data are formatted as follows:

• name: A user-defined name for the experiment, used for naming saved figures.
• dimensions: The dimensions of the data manifold.
• noise: The amount of Gaussian noise to add, in the range [0, 1].
• n_samples: The number of samples in the dataset.
• seed: A random seed.
• datatype: The type of manifold. Refer to generate_data.py to see possible data types.

The params files for cryo-EM data are formatted as follows:

• name: A user-defined name for the experiment, used for naming saved figures.
• var: The variance of Gaussian noise to add to the images.
• n_samples: The number of samples in the dataset.
• seed: A random seed.
• x_stretch: The range [-x, x] which the stretching subunit can stretch in the x-direction.
• y_stretch: The range [-y, y] which the stretching subunit can stretch in the y-direction.
• datatype: The type of manifold, must be set to "cryo-em".

To run the algorithm on a particular dataset, use

python run_experiments.py <path/to/data.pkl> <path/to/configs.json> --generate_plots

For example,

python run_experiments.py data/rectangle3d_info.pkl configs/configs_rectangle3d.json --generate_plots

will run the algorithm on the dataset specified in data/rectangle3d_info.json and generate figures for the different experiments. To run the experiments without producing figures, simply omit the --generate_plots flag.

The settings for the algorithm used in the experiments from the paper are stored in a .json file.

The format of parameters is:

• sigma: The width of the kernel for constructing the data graph.
• n_factors: The desired number of factors to extract.
• n_eigenvectors: The number of eigenvectors to compute.
• eig_crit: The threshold for the eigenvalue criterion (see Section 3.1 of the paper for more details).
• sim_crit: The threshold for the similarity criterion (see Section 3.1 of the paper for more details).
• K: The voting threshold.
• seed: A random seed.
• uniform: Set to true if the data was uniformly sampled, otherwise set to false.

To reproduce the figures in the paper, simply run the script ./reproduce_results.sh.

Alternatively, you can perform the following steps.

First, generate the data:

python generate_data.py \
  params/params_rectangle3d.json \
  params/params_torus.json \
  params/params_cryo-em_x-theta_noisy.json \
  params/params_cube.json

Then, run the experiments:

python factorize.py --data data/rectangle3d_info.pkl --configs configs/configs_rectangle3d.json --outdir results/rectangle3d

python factorize.py --data data/torus_info.pkl --configs configs/configs_torus.json --outdir results/torus

python factorize.py --data data/cryo-em_x-theta_noisy_info.pkl --configs configs/configs_cryo-em_x-theta_noisy.json --outdir results/cryo-em_x-theta_noisy

python factorize.py --data data/cube_info.pkl --configs configs/configs_cube.json --outdir results/cube

And then run the following:

python reproduce_figures.py

The figures will be located in the figures/ folder.