POT: Python Optimal Transport

Contents

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] and stabilized version [9][10] with optional GPU implementation (required cudamat).
  • 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].
  • Joint OT matrix and mapping estimation [8].
  • Wasserstein Discriminant Analysis [11] (requires autograd + pymanopt).
  • Gromov-Wasserstein distances and barycenters [12]

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

Installation

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

  • Numpy (>=1.11)
  • Scipy (>=0.17)
  • Cython (>=0.23)
  • Matplotlib (>=1.5)

Pip installation

You can install the toolbox through PyPI with:

pip install POT

or get the very latest version by downloading it and then running:

python setup.py install --user # for user install (no root)

Anaconda installation with conda-forge

If you use the Anaconda python distribution, POT is available in conda-forge. To install it and the required dependencies:

conda install -c conda-forge pot

Post installation check

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.

Dependencies

Some sub-modules require additional dependences which are discussed below

  • ot.dr (Wasserstein dimensionality rediuction) depends on autograd and pymanopt that can be installed with:

    pip install pymanopt autograd
    
  • ot.gpu (GPU accelerated OT) depends on cudamat that have to be installed with:

    git clone https://github.com/cudamat/cudamat.git
    cd cudamat
    python setup.py install --user # for user install (no root)
    

obviously you need CUDA installed and a compatible GPU.

Examples

  • Import the toolbox

    import ot
    
  • Compute Wasserstein distances

    # a,b are 1D histograms (sum to 1 and positive)
    # M is the ground cost matrix
    Wd=ot.emd2(a,b,M) # exact linear program
    Wd_reg=ot.sinkhorn2(a,b,M,reg) # entropic regularized OT
    # if b is a matrix compute all distances to a and return a vector
    
  • Compute OT matrix

    # a,b are 1D histograms (sum to 1 and positive)
    # M is the ground cost matrix
    T=ot.emd(a,b,M) # exact linear program
    T_reg=ot.sinkhorn(a,b,M,reg) # entropic regularized OT
    
  • Compute Wasserstein barycenter

    # A is a n*d matrix containing d  1D histograms
    # M is the ground cost matrix
    ba=ot.barycenter(A,M,reg) # reg is regularization parameter
    

The examples folder contain several examples and use case for the library. The full documentation is available on Readthedocs.

Here is a list of the Python notebooks available here if you want a quick look:

You can also see the notebooks with Jupyter nbviewer.

Acknowledgements

The contributors to this library are:

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):

Contributions and code of conduct

Every contribution is welcome and should respect the contribution guidelines. Each member of the project is expected to follow the code of conduct.

Support

You can ask questions and join the development discussion:

You can also post bug reports and feature requests in Github issues. Make sure to read our guidelines first.

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.

[8] M. Perrot, N. Courty, R. Flamary, A. Habrard, Mapping estimation for discrete optimal transport, Neural Information Processing Systems (NIPS), 2016.

[9] Schmitzer, B. (2016). Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems. arXiv preprint arXiv:1610.06519.

[10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.

[11] Flamary, R., Cuturi, M., Courty, N., & Rakotomamonjy, A. (2016). Wasserstein Discriminant Analysis. arXiv preprint arXiv:1608.08063.

[12] Gabriel Peyré, Marco Cuturi, and Justin Solomon, Gromov-Wasserstein averaging of kernel and distance matrices International Conference on Machine Learning (ICML). 2016.

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