# POT: Python Optimal Transport

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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/](https://PythonOT.github.io/)

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

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

* [OT Network Simplex solver](https://pythonot.github.io/auto_examples/plot_OT_1D.html) for the linear program/ Earth Movers Distance [1] .
* [Conditional gradient](https://pythonot.github.io/auto_examples/plot_optim_OTreg.html) [6] and [Generalized conditional gradient](https://pythonot.github.io/auto_examples/plot_optim_OTreg.html) for regularized OT [7].
* Entropic regularization OT solver with [Sinkhorn Knopp Algorithm](https://pythonot.github.io/auto_examples/plot_OT_1D.html) [2] , stabilized version [9] [10], greedy Sinkhorn [22] and [Screening Sinkhorn [26] ](https://pythonot.github.io/auto_examples/plot_screenkhorn_1D.html) with optional GPU implementation (requires cupy).
* Bregman projections for [Wasserstein barycenter](https://pythonot.github.io/auto_examples/barycenters/plot_barycenter_lp_vs_entropic.html) [3], [convolutional barycenter](https://pythonot.github.io/auto_examples/barycenters/plot_convolutional_barycenter.html) [21]  and unmixing [4].
* Sinkhorn divergence [23] and entropic regularization OT  from empirical data.
* [Smooth optimal transport solvers](https://pythonot.github.io/auto_examples/plot_OT_1D_smooth.html) (dual and semi-dual) for KL and squared L2 regularizations [17].
* Non regularized [Wasserstein barycenters [16] ](https://pythonot.github.io/auto_examples/barycenters/plot_barycenter_lp_vs_entropic.html)) with LP solver (only small scale).
* [Gromov-Wasserstein distances](https://pythonot.github.io/auto_examples/gromov/plot_gromov.html) and [GW barycenters](https://pythonot.github.io/auto_examples/gromov/plot_gromov_barycenter.html)  (exact [13] and regularized [12])
 * [Fused-Gromov-Wasserstein distances solver](https://pythonot.github.io/auto_examples/gromov/plot_fgw.html#sphx-glr-auto-examples-plot-fgw-py) and [FGW barycenters](https://pythonot.github.io/auto_examples/gromov/plot_barycenter_fgw.html) [24]
* [Stochastic solver](https://pythonot.github.io/auto_examples/plot_stochastic.html) for Large-scale Optimal Transport (semi-dual problem [18] and dual problem [19])
* Non regularized [free support Wasserstein barycenters](https://pythonot.github.io/auto_examples/barycenters/plot_free_support_barycenter.html) [20].
* [Unbalanced OT](https://pythonot.github.io/auto_examples/unbalanced-partial/plot_UOT_1D.html) with KL relaxation and [barycenter](https://pythonot.github.io/auto_examples/unbalanced-partial/plot_UOT_barycenter_1D.html) [10, 25].
* [Partial Wasserstein and Gromov-Wasserstein](https://pythonot.github.io/auto_examples/unbalanced-partial/plot_partial_wass_and_gromov.html) (exact [29] and entropic [3]
  formulations).

POT provides the following Machine Learning related solvers:

* [Optimal transport for domain
  adaptation](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_classes.html)
  with [group lasso regularization](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_classes.html),   [Laplacian regularization](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_laplacian.html) [5] [30] and [semi
  supervised setting](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_semi_supervised.html).
* [Linear OT mapping](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_linear_mapping.html) [14] and [Joint OT mapping estimation](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_mapping.html) [8].
* [Wasserstein Discriminant Analysis](https://pythonot.github.io/auto_examples/others/plot_WDA.html) [11] (requires autograd + pymanopt).
* [JCPOT algorithm for multi-source domain adaptation with target shift](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_jcpot.html) [27].

Some other examples are available in the  [documentation](https://pythonot.github.io/auto_examples/index.html).

#### Using and citing the toolbox

If you use this toolbox in your research and find it useful, please cite POT
using the following reference:
```
Rémi Flamary and Nicolas Courty, POT Python Optimal Transport library, 
Website: https://pythonot.github.io/, 2017
```

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

## 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](https://conda-forge.org). 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:
```python
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](https://docs-cupy.chainer.org/en/stable/install.html).


obviously you need CUDA installed and a compatible GPU.

## Examples

### Short examples

* Import the toolbox
```python
import ot
```
* Compute Wasserstein distances
```python
# 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
```python
# 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
```python
# 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 with examples and output is available on [https://PythonOT.github.io/](https://PythonOT.github.io/).


## Acknowledgements

This toolbox has been created and is maintained by

* [Rémi Flamary](http://remi.flamary.com/)
* [Nicolas Courty](http://people.irisa.fr/Nicolas.Courty/)

The contributors to this library are 

* [Alexandre Gramfort](http://alexandre.gramfort.net/) (CI, documentation)
* [Laetitia Chapel](http://people.irisa.fr/Laetitia.Chapel/) (Partial OT)
* [Michael Perrot](http://perso.univ-st-etienne.fr/pem82055/) (Mapping estimation)
* [Léo Gautheron](https://github.com/aje) (GPU implementation)
* [Nathalie Gayraud](https://www.linkedin.com/in/nathalie-t-h-gayraud/?ppe=1) (DA classes)
* [Stanislas Chambon](https://slasnista.github.io/) (DA classes)
* [Antoine Rolet](https://arolet.github.io/) (EMD solver debug)
* Erwan Vautier (Gromov-Wasserstein)
* [Kilian Fatras](https://kilianfatras.github.io/) (Stochastic solvers)
* [Alain Rakotomamonjy](https://sites.google.com/site/alainrakotomamonjy/home)
* [Vayer Titouan](https://tvayer.github.io/) (Gromov-Wasserstein -, Fused-Gromov-Wasserstein)
* [Hicham Janati](https://hichamjanati.github.io/) (Unbalanced OT)
* [Romain Tavenard](https://rtavenar.github.io/) (1d Wasserstein)
* [Mokhtar Z. Alaya](http://mzalaya.github.io/) (Screenkhorn)
* [Ievgen Redko](https://ievred.github.io/) (Laplacian DA, JCPOT)

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](http://gpeyre.github.io/) (Wasserstein Barycenters in Matlab)
* [Nicolas Bonneel](http://liris.cnrs.fr/~nbonneel/) ( C++ code for EMD)
* [Marco Cuturi](http://marcocuturi.net/) (Sinkhorn Knopp in Matlab/Cuda)


## Contributions and code of conduct

Every contribution is welcome and should respect the [contribution guidelines](CONTRIBUTING.md). Each member of the project is expected to follow the [code of conduct](CODE_OF_CONDUCT.md).

## Support

You can ask questions and join the development discussion:

* On the [POT Slack channel](https://pot-toolbox.slack.com)
* On the POT [mailing list](https://mail.python.org/mm3/mailman3/lists/pot.python.org/)


You can also post bug reports and feature requests in Github issues. Make sure to read our [guidelines](CONTRIBUTING.md) first.

## References

[1] Bonneel, N., Van De Panne, M., Paris, S., & Heidrich, W. (2011, December). [Displacement interpolation using Lagrangian mass transport](https://people.csail.mit.edu/sparis/publi/2011/sigasia/Bonneel_11_Displacement_Interpolation.pdf). In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p. 158). ACM.

[2] Cuturi, M. (2013). [Sinkhorn distances: Lightspeed computation of optimal transport](https://arxiv.org/pdf/1306.0895.pdf). 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](https://arxiv.org/pdf/1412.5154.pdf). 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](https://hal.archives-ouvertes.fr/hal-01377236/document), 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](https://arxiv.org/pdf/1507.00504.pdf), 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](https://arxiv.org/pdf/1307.5551.pdf). 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](https://arxiv.org/pdf/1510.06567.pdf). arXiv preprint arXiv:1510.06567.

[8] M. Perrot, N. Courty, R. Flamary, A. Habrard (2016), [Mapping estimation for discrete optimal transport](http://remi.flamary.com/biblio/perrot2016mapping.pdf), Neural Information Processing Systems (NIPS).

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

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

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

[12] Gabriel Peyré, Marco Cuturi, and Justin Solomon (2016), [Gromov-Wasserstein averaging of kernel and distance matrices](http://proceedings.mlr.press/v48/peyre16.html)  International Conference on Machine Learning (ICML).

[13] Mémoli, Facundo (2011). [Gromov–Wasserstein distances and the metric approach to object matching](https://media.adelaide.edu.au/acvt/Publications/2011/2011-Gromov%E2%80%93Wasserstein%20Distances%20and%20the%20Metric%20Approach%20to%20Object%20Matching.pdf). 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](https://arxiv.org/pdf/1803.00567.pdf) .

[16] Agueh, M., & Carlier, G. (2011). [Barycenters in the Wasserstein space](https://hal.archives-ouvertes.fr/hal-00637399/document). SIAM Journal on Mathematical Analysis, 43(2), 904-924.

[17] Blondel, M., Seguy, V., & Rolet, A. (2018). [Smooth and Sparse Optimal Transport](https://arxiv.org/abs/1710.06276). 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](https://arxiv.org/abs/1605.08527). 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](https://arxiv.org/pdf/1711.02283.pdf). International Conference on Learning Representation (2018)

[20] Cuturi, M. and Doucet, A. (2014) [Fast Computation of Wasserstein Barycenters](http://proceedings.mlr.press/v32/cuturi14.html). 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](https://dl.acm.org/citation.cfm?id=2766963). 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](https://papers.nips.cc/paper/6792-near-linear-time-approximation-algorithms-for-optimal-transport-via-sinkhorn-iteration.pdf), Advances in Neural Information Processing Systems (NIPS) 31

[23] Aude, G., Peyré, G., Cuturi, M., [Learning Generative Models with Sinkhorn Divergences](https://arxiv.org/abs/1706.00292), 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](http://proceedings.mlr.press/v97/titouan19a.html) 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](http://cbcl.mit.edu/wasserstein/)  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](https://papers.nips.cc/paper/9386-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](http://proceedings.mlr.press/v89/redko19a.html), 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](http://www.math.toronto.edu/~mccann/papers/annals2010.pdf), Annals of mathematics, 673-730.

[29] Chapel, L., Alaya, M., Gasso, G. (2019). [Partial Gromov-Wasserstein with Applications on Positive-Unlabeled Learning](https://arxiv.org/abs/2002.08276), arXiv preprint arXiv:2002.08276.

[30] Flamary R., Courty N., Tuia D., Rakotomamonjy A. (2014). [Optimal transport with Laplacian regularization: Applications to domain adaptation and shape matching](https://remi.flamary.com/biblio/flamary2014optlaplace.pdf), NIPS Workshop on Optimal Transport and Machine Learning OTML, 2014.