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/

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

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

-  `OT Network Simplex
   solver <auto_examples/plot_OT_1D.html>`__
   for the linear program/ Earth Movers Distance [1] .
-  `Conditional
   gradient <auto_examples/plot_optim_OTreg.html>`__
   [6] and `Generalized conditional
   gradient <auto_examples/plot_optim_OTreg.html>`__
   for regularized OT [7].
-  Entropic regularization OT solver with `Sinkhorn Knopp
   Algorithm <auto_examples/plot_OT_1D.html>`__
   [2] , stabilized version [9] [10], greedy Sinkhorn [22] and
   `Screening Sinkhorn
   [26] <auto_examples/plot_screenkhorn_1D.html>`__
   with optional GPU implementation (requires cupy).
-  Bregman projections for `Wasserstein
   barycenter <auto_examples/barycenters/plot_barycenter_lp_vs_entropic.html>`__
   [3], `convolutional
   barycenter <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 <auto_examples/plot_OT_1D_smooth.html>`__
   (dual and semi-dual) for KL and squared L2 regularizations [17].
-  Non regularized `Wasserstein barycenters
   [16] <auto_examples/barycenters/plot_barycenter_lp_vs_entropic.html>`__)
   with LP solver (only small scale).
-  `Gromov-Wasserstein
   distances <auto_examples/gromov/plot_gromov.html>`__
   and `GW
   barycenters <auto_examples/gromov/plot_gromov_barycenter.html>`__
   (exact [13] and regularized [12])
-  `Fused-Gromov-Wasserstein distances
   solver <auto_examples/gromov/plot_fgw.html#sphx-glr-auto-examples-plot-fgw-py>`__
   and `FGW
   barycenters <auto_examples/gromov/plot_barycenter_fgw.html>`__
   [24]
-  `Stochastic
   solver <auto_examples/plot_stochastic.html>`__
   for Large-scale Optimal Transport (semi-dual problem [18] and dual
   problem [19])
-  Non regularized `free support Wasserstein
   barycenters <auto_examples/barycenters/plot_free_support_barycenter.html>`__
   [20].
-  `Unbalanced
   OT <auto_examples/unbalanced-partial/plot_UOT_1D.html>`__
   with KL relaxation and
   `barycenter <auto_examples/unbalanced-partial/plot_UOT_barycenter_1D.html>`__
   [10, 25].
-  `Partial Wasserstein and
   Gromov-Wasserstein <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 <auto_examples/domain-adaptation/plot_otda_classes.html>`__
   with `group lasso
   regularization <auto_examples/domain-adaptation/plot_otda_classes.html>`__,
   `Laplacian
   regularization <auto_examples/domain-adaptation/plot_otda_laplacian.html>`__
   [5] [30] and `semi supervised
   setting <auto_examples/domain-adaptation/plot_otda_semi_supervised.html>`__.
-  `Linear OT
   mapping <auto_examples/domain-adaptation/plot_otda_linear_mapping.html>`__
   [14] and `Joint OT mapping
   estimation <auto_examples/domain-adaptation/plot_otda_mapping.html>`__
   [8].
-  `Wasserstein Discriminant
   Analysis <auto_examples/others/plot_WDA.html>`__
   [11] (requires autograd + pymanopt).
-  `JCPOT algorithm for multi-source domain adaptation with target
   shift <auto_examples/domain-adaptation/plot_otda_jcpot.html>`__
   [27].

Some other examples are available in the
`documentation <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:

.. code:: 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

   .. code:: python

       import ot

-  Compute Wasserstein distances

   .. code:: 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

   .. code:: 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

   .. code:: 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/.

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.

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