# 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 Network Flow solver for the linear program/ Earth Movers Distance [1].
- Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2], stabilized version [9][10] and greedy Sinkhorn [22] with optional GPU implementation (requires cupy).
- Sinkhorn divergence [23] and entropic regularization OT from empirical data.
- Smooth optimal transport solvers (dual and semi-dual) for KL and squared L2 regularizations [17].
- Non regularized Wasserstein barycenters [16] with LP solver (only small scale).
- Bregman projections for Wasserstein barycenter [3], convolutional barycenter [21] and unmixing [4].
- Optimal transport for domain adaptation with group lasso regularization [5]
- Conditional gradient [6] and Generalized conditional gradient for regularized OT [7].
- Linear OT [14] and Joint OT matrix and mapping estimation [8].
- Wasserstein Discriminant Analysis [11] (requires autograd + pymanopt).
- Gromov-Wasserstein distances and barycenters ([13] and regularized [12])
- Stochastic Optimization for Large-scale Optimal Transport (semi-dual problem [18] and dual problem [19])
- Non regularized free support Wasserstein barycenters [20].
- Unbalanced OT with KL relaxation distance and barycenter [10, 25].

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

### Using and citing the toolbox¶

If you use this toolbox in your research and find it useful, please cite POT using the following bibtex reference:

```
@misc{flamary2017pot,
title={POT Python Optimal Transport library},
author={Flamary, R{'e}mi and Courty, Nicolas},
url={https://github.com/rflamary/POT},
year={2017}
}
```

## 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 (>=1.0)
- Cython (>=0.23)
- Matplotlib (>=1.5)

### Pip installation¶

Note that due to a limitation of pip, `cython`

and `numpy`

need to
be installed prior to installing POT. This can be done easily with

```
pip install numpy cython
```

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 reduction) depends on autograd and pymanopt that can be installed with:pip install pymanopt autograd

**ot.gpu**(GPU accelerated OT) depends on cupy that have to be installed following instructions on this page.

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:

- 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
- Gromov Wasserstein
- Gromov Wasserstein Barycenter

You can also see the notebooks with Jupyter nbviewer.

## Acknowledgements¶

This toolbox has been created and is maintained by

The contributors to this library are

- Alexandre Gramfort
- Laetitia Chapel
- Michael Perrot (Mapping estimation)
- Léo Gautheron (GPU implementation)
- Nathalie Gayraud
- Stanislas Chambon
- Antoine Rolet
- Erwan Vautier (Gromov-Wasserstein)
- Kilian Fatras
- Alain Rakotomamonjy
- Vayer Titouan
- Hicham Janati (Unbalanced OT)
- Romain Tavenard (1d Wasserstein)

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)
- Marco Cuturi (Sinkhorn Knopp in Matlab/Cuda)

## 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:

- On the POT Slack channel
- On the POT mailing list

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 (2016), Mapping estimation for discrete optimal transport, Neural Information Processing Systems (NIPS).

[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 (2016), Gromov-Wasserstein averaging of kernel and distance matrices International Conference on Machine Learning (ICML).

[13] Mémoli, Facundo (2011). Gromov–Wasserstein distances and the metric approach to object matching. Foundations of computational mathematics 11.4 : 417-487.

[14] Knott, M. and Smith, C. S. (1984).`On the optimal mapping of distributions <https://link.springer.com/article/10.1007/BF00934745>`__, Journal of Optimization Theory and Applications Vol 43.

[15] Peyré, G., & Cuturi, M. (2018). Computational Optimal Transport .

[16] Agueh, M., & Carlier, G. (2011). Barycenters in the Wasserstein space. SIAM Journal on Mathematical Analysis, 43(2), 904-924.

[17] Blondel, M., Seguy, V., & Rolet, A. (2018). Smooth and Sparse Optimal Transport. Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics (AISTATS).

[18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016) Stochastic Optimization for Large-scale Optimal Transport. Advances in Neural Information Processing Systems (2016).

[19] Seguy, V., Bhushan Damodaran, B., Flamary, R., Courty, N., Rolet, A.& Blondel, M. Large-scale Optimal Transport and Mapping Estimation. International Conference on Learning Representation (2018)

[20] Cuturi, M. and Doucet, A. (2014) Fast Computation of Wasserstein Barycenters. International Conference in Machine Learning

[21] Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher, A., Nguyen, A. & Guibas, L. (2015). Convolutional wasserstein distances: Efficient optimal transportation on geometric domains. ACM Transactions on Graphics (TOG), 34(4), 66.

[22] J. Altschuler, J.Weed, P. Rigollet, (2017) Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration, Advances in Neural Information Processing Systems (NIPS) 31

[23] Aude, G., Peyré, G., Cuturi, M., Learning Generative Models with Sinkhorn Divergences, Proceedings of the Twenty-First International Conference on Artficial Intelligence and Statistics, (AISTATS) 21, 2018

[24] Vayer, T., Chapel, L., Flamary, R., Tavenard, R. and Courty, N. (2019). Optimal Transport for structured data with application on graphs Proceedings of the 36th International Conference on Machine Learning (ICML).

[25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. (2019). Learning with a Wasserstein Loss Advances in Neural Information Processing Systems (NIPS).