Welcome to ot_markov_distances’s documentation!


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Differentiable distances on graphs based on optimal transport

This is the implementation code for

Brugere, T., Wan, Z., & Wang, Y. (2023). Distances for Markov Chains, and Their Differentiation. ArXiv, abs/2302.08621.


Installing as a library

The ot_markov_distances package can be installed with the following command:

pip install ot-markov-distances

If for some reason you need to use cuda11.8 (ie you are installing torch+cuda118) then use the following command instead

pip install git+https://github.com/YusuLab/ot_markov_distances@cuda118


Python version

This project requires python 3.10 a minima. If your python version is prior to 3.10, you need to update (or to create a new conda environment) to a version above (latest release at the time of writing is 3.12)

Python dependencies


The main branch uses the default (cuda12) version of torch in its dependencies. If for some reason you need to use cuda11.8, clone the cuda118 branch instead

This package manages its dependencies via poetry. I recommend you install it (otherwise if you prefer to manage them manually, a list of the dependencies is available in the file pyproject.toml)

When you have poetry, you can add dependencies using our makefile


If you want to create a virtual environment for this project (as opposed to using the one you are currently in) you can use the command poetry env use python3.12 (or other python version)

$ make .make/deps

or directly with poetry

$ poetry install


If you are planning to reproduce the classification experiment.

The TUDataset package is also needed to run the classification experiment, but it is not available via pip / poetry. To install it, follow the instruction in the tudataset repo, including the “Compilation of kernel baselines” section, and add the directory where you downloaded it to your $PYTHONPATH. eg:

$ export PYTHONPATH="/path/to/tudataset:$PYTHONPATH"

Project structure

├── docs    #contains the generated docs (after typing make)
│   ├── build
│   │   └── html            #Contains the html docs in readthedocs format
│   └── source
├── experiments             #contains jupyter notebooks with the experiments
│   └── utils               #contains helper code for the experiments
├── ot_markov_distances     #contains reusable library code for computing and differentiating the discounted WL distance
│   ├── discounted_wl.py    # implementation of our discounted WL distance
│   ├── __init__.py
│   ├── sinkhorn.py         # implementation of the sinkhorn distance
│   ├── utils.py            # utility functions
│   └── wl.py               #implementation of the wl distance by Chen et al.
├── staticdocs #contains the static source for the docs
│   ├── build
│   └── source
└── tests #contains sanity checks


The documentation is available online: read the documentation


Do not edit the documentation directly in the docs/ folder, that folder is wiped every time the documentation is built. The static parts of the documentation can be edited in staticdocs/.

You can build documentation and run tests using

$ make

Alternatively, you can build only the documentation using

$ make .make/build-docs

The documentation will be available in docs/build/html in the readthedocs format

Running Experiments

Running experiments requires installing development dependencies. This can be done by running

$ make .make/dev-deps

or alternatively

$ poetry install --with dev

Experiments can be found in the experiments/ directory (see Project structure ).

The Barycenter and Coarsening experiments can be found in experiments/Barycenter.ipynb and experiments/Coarsening.ipynb.

The performance graphs are computed in experiments/Performance.ipynb

Classification experiment

The Classification experiment (see the first paragraph of section 6 in the paper) is not in a jupyter notebook, but accessible via a command line.

As an additional dependency it needs tudataset, which is not installable via pip. To install it follow the instructions in the tudataset repo. , including the “Compilation of kernel baselines” section, and add the directory where you downloaded it to your $PYTHONPATH.

Now you can run the classification experiment using the command

$ poetry run python -m experiments.classification
usage: python -m experiments.classification [-h] {datasets_info,distances,eval} ...

Run classification experiments on graph datasets

positional arguments:
    datasets_info       Print information about given datasets
    distances           Compute distance matrices for given datasets
    eval                Evaluate a kernel based on distance matrix

  -h, --help            show this help message and exit

The yaml file containing dataset information that should be passed to the command line is in experiments/grakel_datasets.yaml. Modifying this file should allow running the experiment on different datasets.


I have a question about the paper

In this case just send me an email through the email address mentioned in the paper.

I have noticed a bug in the code

Please use the Github “Issues” feature to open a ticket, and post a description of the bug, the error message and a minimal reproducible example . I’ll try to fix it.

Or if you have fixed it, you can submit a Pull Request directly

I cannot install the library

If you followed all the instructions correctly, please create a ticket using Github Issues.

Why do you need python3.10 ?

Because I am using structural pattern matching, and some typing features such as this one .

Indices and tables



Tristan Brugère, Zhengchao Wan, and Yusu Wang. Distances for markov chains, and their differentiation. 2023. arXiv:2302.08621.


Samantha Chen, Sunhyuk Lim, Facundo Mémoli, Zhengchao Wan, and Yusu Wang. Weisfeiler-Lehman meets Gromov-Wasserstein. In International Conference on Machine Learning (ICML), 3371–3416. PMLR, 2022.


Jean Feydy, Thibault Séjourné, François-Xavier Vialard, Shun-ichi Amari, Alain Trouvé, and Gabriel Peyré. Interpolating between optimal transport and mmd using sinkhorn divergences. In The 22nd International Conference on Artificial Intelligence and Statistics, 2681–2690. PMLR, 2019.