Python Optimal Transport Library

## Project description

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.

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