# POT: Python Optimal Transport [![PyPI version](https://badge.fury.io/py/POT.svg)](https://badge.fury.io/py/POT) [![Anaconda Cloud](https://anaconda.org/conda-forge/pot/badges/version.svg)](https://anaconda.org/conda-forge/pot) [![Build Status](https://travis-ci.org/rflamary/POT.svg?branch=master)](https://travis-ci.org/rflamary/POT) [![Documentation Status](https://readthedocs.org/projects/pot/badge/?version=latest)](http://pot.readthedocs.io/en/latest/?badge=latest) [![Downloads](https://pepy.tech/badge/pot)](https://pepy.tech/project/pot) [![Anaconda downloads](https://anaconda.org/conda-forge/pot/badges/downloads.svg)](https://anaconda.org/conda-forge/pot) [![License](https://anaconda.org/conda-forge/pot/badges/license.svg)](https://github.com/rflamary/POT/blob/master/LICENSE) 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 building/installing 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](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 is available on [Readthedocs](http://pot.readthedocs.io/). Here is a list of the Python notebooks available [here](https://github.com/rflamary/POT/blob/master/notebooks/) if you want a quick look: * [1D optimal transport](https://github.com/rflamary/POT/blob/master/notebooks/plot_OT_1D.ipynb) * [OT Ground Loss](https://github.com/rflamary/POT/blob/master/notebooks/plot_OT_L1_vs_L2.ipynb) * [Multiple EMD computation](https://github.com/rflamary/POT/blob/master/notebooks/plot_compute_emd.ipynb) * [2D optimal transport on empirical distributions](https://github.com/rflamary/POT/blob/master/notebooks/plot_OT_2D_samples.ipynb) * [1D Wasserstein barycenter](https://github.com/rflamary/POT/blob/master/notebooks/plot_barycenter_1D.ipynb) * [OT with user provided regularization](https://github.com/rflamary/POT/blob/master/notebooks/plot_optim_OTreg.ipynb) * [Domain adaptation with optimal transport](https://github.com/rflamary/POT/blob/master/notebooks/plot_otda_d2.ipynb) * [Color transfer in images](https://github.com/rflamary/POT/blob/master/notebooks/plot_otda_color_images.ipynb) * [OT mapping estimation for domain adaptation](https://github.com/rflamary/POT/blob/master/notebooks/plot_otda_mapping.ipynb) * [OT mapping estimation for color transfer in images](https://github.com/rflamary/POT/blob/master/notebooks/plot_otda_mapping_colors_images.ipynb) * [Wasserstein Discriminant Analysis](https://github.com/rflamary/POT/blob/master/notebooks/plot_WDA.ipynb) * [Gromov Wasserstein](https://github.com/rflamary/POT/blob/master/notebooks/plot_gromov.ipynb) * [Gromov Wasserstein Barycenter](https://github.com/rflamary/POT/blob/master/notebooks/plot_gromov_barycenter.ipynb) * [Fused Gromov Wasserstein](https://github.com/rflamary/POT/blob/master/notebooks/plot_fgw.ipynb) * [Fused Gromov Wasserstein Barycenter](https://github.com/rflamary/POT/blob/master/notebooks/plot_barycenter_fgw.ipynb) You can also see the notebooks with [Jupyter nbviewer](https://nbviewer.jupyter.org/github/rflamary/POT/tree/master/notebooks/). ## 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/) * [Laetitia Chapel](http://people.irisa.fr/Laetitia.Chapel/) * [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) * [Stanislas Chambon](https://slasnista.github.io/) * [Antoine Rolet](https://arolet.github.io/) * Erwan Vautier (Gromov-Wasserstein) * [Kilian Fatras](https://kilianfatras.github.io/) * [Alain Rakotomamonjy](https://sites.google.com/site/alainrakotomamonjy/home) * [Vayer Titouan](https://tvayer.github.io/) * [Hicham Janati](https://hichamjanati.github.io/) (Unbalanced OT) * [Romain Tavenard](https://rtavenar.github.io/) (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](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. (2019). [Learning with a Wasserstein Loss](http://cbcl.mit.edu/wasserstein/) Advances in Neural Information Processing Systems (NIPS).