# 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.

Website and documentation: https://PythonOT.github.io/

Source Code (MIT): https://github.com/PythonOT/POT

POT provides the following generic OT solvers (links to examples):

OT Network Simplex solver for the linear program/ Earth Movers Distance [1] .

Conditional gradient [6] and Generalized conditional gradient for regularized OT [7].

Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2] , stabilized version [9] [10], greedy Sinkhorn [22] and Screening Sinkhorn [26] with optional GPU implementation (requires cupy).

Bregman projections for Wasserstein barycenter [3], convolutional barycenter [21] and unmixing [4].

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).

Gromov-Wasserstein distances and GW barycenters (exact [13] and regularized [12])

Fused-Gromov-Wasserstein distances solver and FGW barycenters [24]

Stochastic solver 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 and barycenter [10, 25].

Partial Wasserstein and Gromov-Wasserstein (exact [29] and entropic [3] formulations).

Sliced Wasserstein [31, 32].

POT provides the following Machine Learning related solvers:

Optimal transport for domain adaptation with group lasso regularization, Laplacian regularization [5] [30] and semi supervised setting.

Linear OT mapping [14] and Joint OT mapping estimation [8].

Wasserstein Discriminant Analysis [11] (requires autograd + pymanopt).

JCPOT algorithm for multi-source domain adaptation with target shift [27].

Some other examples are available in the documentation.

### Using and citing the toolbox¶

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

```
Rémi Flamary, Nicolas Courty, Alexandre Gramfort, Mokhtar Z. Alaya, Aurélie Boisbunon, Stanislas Chambon, Laetitia Chapel, Adrien Corenflos, Kilian Fatras, Nemo Fournier, Léo Gautheron, Nathalie T.H. Gayraud, Hicham Janati, Alain Rakotomamonjy, Ievgen Redko, Antoine Rolet, Antony Schutz, Vivien Seguy, Danica J. Sutherland, Romain Tavenard, Alexander Tong, Titouan Vayer;, POT Python Optimal Transport library, Journal of Machine Learning Research, 22(78):1−8, 2021.
Website: https://pythonot.github.io/
```

In Bibtex format:

```
@article{flamary2021pot,
author = {R{\'e}mi Flamary and Nicolas Courty and Alexandre Gramfort and Mokhtar Z. Alaya and Aur{\'e}lie Boisbunon and Stanislas Chambon and Laetitia Chapel and Adrien Corenflos and Kilian Fatras and Nemo Fournier and L{\'e}o Gautheron and Nathalie T.H. Gayraud and Hicham Janati and Alain Rakotomamonjy and Ievgen Redko and Antoine Rolet and Antony Schutz and Vivien Seguy and Danica J. Sutherland and Romain Tavenard and Alexander Tong and Titouan Vayer},
title = {POT: Python Optimal Transport},
journal = {Journal of Machine Learning Research},
year = {2021},
volume = {22},
number = {78},
pages = {1-8},
url = {http://jmlr.org/papers/v22/20-451.html}
}
```

## Installation¶

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

Numpy (>=1.16)

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 running:

```
pip install -U https://github.com/PythonOT/POT/archive/master.zip # with --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 with examples and output is available on https://PythonOT.github.io/.

## Acknowledgements¶

This toolbox has been created and is maintained by

The contributors to this library are

Alexandre Gramfort (CI, documentation)

Laetitia Chapel (Partial OT)

Michael Perrot (Mapping estimation)

Léo Gautheron (GPU implementation)

Nathalie Gayraud (DA classes)

Stanislas Chambon (DA classes)

Antoine Rolet (EMD solver debug)

Erwan Vautier (Gromov-Wasserstein)

Kilian Fatras (Stochastic solvers)

Vayer Titouan (Gromov-Wasserstein -, Fused-Gromov-Wasserstein)

Hicham Janati (Unbalanced OT)

Romain Tavenard (1d Wasserstein)

Mokhtar Z. Alaya (Screenkhorn)

Ievgen Redko (Laplacian DA, JCPOT)

Adrien Corenflos (Sliced Wasserstein Distance)

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)

Mathieu Blondel (original implementation smooth OT)

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 gitter 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. (2015). Learning with a Wasserstein Loss Advances in Neural Information Processing Systems (NIPS).

[26] Alaya M. Z., Bérar M., Gasso G., Rakotomamonjy A. (2019). Screening Sinkhorn Algorithm for Regularized Optimal Transport, Advances in Neural Information Processing Systems 33 (NeurIPS).

[27] Redko I., Courty N., Flamary R., Tuia D. (2019). Optimal Transport for Multi-source Domain Adaptation under Target Shift, Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics (AISTATS) 22, 2019.

[28] Caffarelli, L. A., McCann, R. J. (2010). Free boundaries in optimal transport and Monge-Ampere obstacle problems, Annals of mathematics, 673-730.

[29] Chapel, L., Alaya, M., Gasso, G. (2020). Partial Optimal Transport with Applications on Positive-Unlabeled Learning, Advances in Neural Information Processing Systems (NeurIPS), 2020.

[30] Flamary R., Courty N., Tuia D., Rakotomamonjy A. (2014). Optimal transport with Laplacian regularization: Applications to domain adaptation and shape matching, NIPS Workshop on Optimal Transport and Machine Learning OTML, 2014.

[31] Bonneel, Nicolas, et al. Sliced and radon wasserstein barycenters of measures, Journal of Mathematical Imaging and Vision 51.1 (2015): 22-45