POT: Python Optimal Transport

This open source Python library provide several solvers for optimization problems related to Optimal Transport for signal, image processing and machine learning.

It provides the following solvers:

• OT solver for the linear program/ Earth Movers Distance [1].
• Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2].
• Bregman projections for Wasserstein barycenter [3] and unmixing [4].
• Optimal transport for domain adaptation with group lasso regularization [5]
• Conditional gradient [6] and Generalized conditional gradient for regularized OT [7].

Some demonstrations (both in Python and Jupyter Notebook format) are available in the examples folder.

Installation

The Library has been tested on Linux and MacOSX. It requires a C++ compiler for using the EMD solver and rely on the following Python modules:

• Numpy (>=1.11)
• Scipy (>=0.17)

To install the library, you can install it locally (after downloading it) on you machine using

python setup.py install --user

The toolbox is also available on PyPI with a possibly slightly older version. You can install it with:

pip install POT

After a correct installation, you should be able to import the module without errors:

import ot

Note that for easier access the module is name ot instead of pot.

Examples

The examples folder contain several examples and use case for the library.

Here is a list of the Python notebook if you want a quick look:

• 1D optimal transport
• 2D optimal transport on empirical distributions
• 1D Wasserstein barycenter
• OT with user provided regularization

Acknowledgements

The contributors to this library are:

• Rémi Flamary
• Nicolas Courty
• Laetitia Chapel

This toolbox benefit a lot from open source research and we would like to thank the following persons for providing some code (in various languages):

• Gabriel Peyré (Wasserstein Barycenters in Matlab)
• Nicolas Bonneel ( C++ code for EMD)
• Antoine Rolet ( Mex file for EMD )
• Marco Cuturi (Sinkhorn Knopp in Matlab/Cuda)

References

[1] Bonneel, N., Van De Panne, M., Paris, S., & Heidrich, W. (2011, December). Displacement interpolation using Lagrangian mass transport. In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p. 158). ACM.

[2] Cuturi, M. (2013). Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems (pp. 2292-2300).

[3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015). Iterative Bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing, 37(2), A1111-A1138.

[4] S. Nakhostin, N. Courty, R. Flamary, D. Tuia, T. Corpetti, Supervised planetary unmixing with optimal transport, Whorkshop on Hyperspectral Image and Signal Processing : Evolution in Remote Sensing (WHISPERS), 2016.

[5] N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, "Optimal Transport for Domain Adaptation," in IEEE Transactions on Pattern Analysis and Machine Intelligence , vol.PP, no.99, pp.1-1

[6] Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014). Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3), 1853-1882.

[7] Rakotomamonjy, A., Flamary, R., & Courty, N. (2015). Generalized conditional gradient: analysis of convergence and applications. arXiv preprint arXiv:1510.06567.