PyVot : Python Variational Optimal Transportation

This package includes the prototype code for computing Monge's optimal transportation (OT) and Wasserstein clustering.

Variational Wasserstein clustering in each iteration leverages variational principles [1] to solve optimal transportation. Thus, we name the package PyVot instead of PyVWc for the sake of pronunciation. For computing optimal transportation, simply set the max iteration to one.
Monge's OT maps exist in general when one of the distributions is absolutely continuous. In practice, we consider a collection of dense Dirac samples as an approximation.
This program implements gradient descent instead of Newton's method to avoid computing convex hulls so that it can handle high-dimensional data.
The picture below shows the results from original Wasserstein clustering [2] and regularized Wasserstein clustering [3]. More examples can be found in demo/

Dependencies

Python >= 3.5
NumPy >= 1.15.4
SciPy >= 1.1.0 (for regularization)
scikit-image >= 0.14.1 (for geometric transformation)
Matplotlib >= 3.0.2 (for demo)

References

Citing the package

If you find the code helpful, please cite one of the following articles:

Mi, Liang, Wen Zhang, Xianfeng Gu, and Yalin Wang. "Variational Wasserstein Clustering." In Proceedings of the European Conference on Computer Vision (ECCV), pp. 322-337. 2018.

@inproceedings{mi2018variational,
  title={Variational {W}asserstein Clustering},
  author={Mi, Liang and Zhang, Wen and Gu, Xianfeng and Wang, Yalin},
  booktitle={Proceedings of the European Conference on Computer Vision (ECCV)},
  pages={322--337},
  year={2018}
}

Mi, Liang, Wen Zhang, and Yalin Wang. "Regularized Wasserstein Means Based on Variational Transportation." arXiv preprint arXiv:1812.00338 (2018).

@article{mi2018regularized,
  title={Regularized Wasserstein Means Based on Variational Transportation},
  author={Mi, Liang and Zhang, Wen and Wang, Yalin},
  journal={arXiv preprint arXiv:1812.00338},
  year={2018}
}

Additional references

[1] Gu, Xianfeng, Feng Luo, Jian Sun, and S-T. Yau. "Variational principles for Minkowski type problems, discrete optimal transport, and discrete Monge-Ampere equations." arXiv preprint arXiv:1302.5472 (2013).

[2] Mi, Liang, Wen Zhang, Xianfeng Gu, and Yalin Wang. "Variational Wasserstein Clustering." In Proceedings of the European Conference on Computer Vision (ECCV), pp. 322-337. 2018.

[3] Mi, Liang, Wen Zhang, and Yalin Wang. "Regularized Wasserstein Means Based on Variational Transportation." arXiv preprint arXiv:1812.00338 (2018).

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