POT 0.3

Python Optimal Transport Library

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).

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)

• Cython (>=0.23)

• Matplotlib (>=1.5)

Under debian based linux the dependencies can be installed with

sudo apt-get install python-numpy python-scipy python-matplotlib cython

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

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

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.

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

Short 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

Examples and Notebooks

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:

• 1D optimal transport

• OT Ground Loss

• Multiple EMD computation

• 2D optimal transport on empirical distributions

• 1D Wasserstein barycenter

• OT with user provided regularization

• Domain adaptation with optimal transport

• Color transfer in images

• OT mapping estimation for domain adaptation

• OT mapping estimation for color transfer in images

• Wasserstein Discriminant Analysis

You can also see the notebooks with Jupyter nbviewer.

Acknowledgements

The contributors to this library are:

• Rémi Flamary

• Nicolas Courty

• Laetitia Chapel

• Michael Perrot (Mapping estimation)

• Léo Gautheron (GPU implementation)

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.

[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.