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] [34], greedy Sinkhorn [22] and
   `Screening Sinkhorn
   [26] <auto_examples/plot_screenkhorn_1D.html>`__.
-  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.
-  Debiased Sinkhorn barycenters `Sinkhorn divergence
   barycenter <auto_examples/barycenters/plot_debiased_barycenter.html>`__
   [37]
-  `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]), differentiable using gradients
   from
-  `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])
-  `Stochastic solver of Gromov
   Wasserstein <auto_examples/gromov/plot_gromov.html>`__
   for large-scale problem with any loss functions [33]
-  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).
-  `Sliced
   Wasserstein <auto_examples/sliced-wasserstein/plot_variance.html>`__
   [31, 32] and Max-sliced Wasserstein [35] that can be used for
   gradient flows [36].
-  `Several
   backends <https://pythonot.github.io/quickstart.html#solving-ot-with-multiple-backends>`__
   for easy use of POT with
   `Pytorch <https://pytorch.org/>`__/`jax <https://github.com/google/jax>`__/`Numpy <https://numpy.org/>`__
   arrays.

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 from our `JMLR
paper <https://jmlr.org/papers/v22/20-451.html>`__:

::

    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:

.. code:: bibtex

    @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) (build only, not necessary when installing from pip
   or conda)

Pip installation
^^^^^^^^^^^^^^^^

You can install the toolbox through PyPI with:

.. code:: console

    pip install POT

or get the very latest version by running:

.. code:: console

    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:

.. code:: console

    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 named ``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:

.. code:: shell

    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 will need CUDA installed and a compatible GPU. Note
   that this module is deprecated since version 0.8 and will be deleted
   in the future. GPU is now handled automatically through the backends
   and several solver already can run on GPU using the Pytorch backend.

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,
   Debiased barycenters)
-  `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)
-  `Adrien Corenflos <https://adriencorenflos.github.io/>`__ (Sliced
   Wasserstein Distance)
-  `Tanguy Kerdoncuff <https://hv0nnus.github.io/>`__ (Sampled Gromov
   Wasserstein)
-  `Minhui Huang <https://mhhuang95.github.io>`__ (Projection Robust
   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 <http://gpeyre.github.io/>`__ (Wasserstein Barycenters
   in Matlab)
-  `Mathieu Blondel <https://mblondel.org/>`__ (original implementation
   smooth OT)
-  `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 <.github/CONTRIBUTING.md>`__. Each member of the project is
expected to follow the `code of conduct <.github/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 `gitter channel <https://gitter.im/PythonOT/community>`__
-  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 <.github/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
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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
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[10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016).
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[11] Flamary, R., Cuturi, M., Courty, N., & Rakotomamonjy, A. (2016).
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[12] Gabriel Peyré, Marco Cuturi, and Justin Solomon (2016),
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International Conference on Machine Learning (ICML).

[13] Mémoli, Facundo (2011). `Gromov–Wasserstein distances and the
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[14] Knott, M. and Smith, C. S. (1984).`On the optimal mapping of
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Journal of Optimization Theory and Applications Vol 43.

[15] Peyré, G., & Cuturi, M. (2018). `Computational Optimal
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[16] Agueh, M., & Carlier, G. (2011). `Barycenters in the Wasserstein
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[17] Blondel, M., Seguy, V., & Rolet, A. (2018). `Smooth and Sparse
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the Twenty-First International Conference on Artificial Intelligence and
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[18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016) `Stochastic
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[19] Seguy, V., Bhushan Damodaran, B., Flamary, R., Courty, N., Rolet,
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[20] Cuturi, M. and Doucet, A. (2014) `Fast Computation of Wasserstein
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[21] Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher, A.,
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[22] J. Altschuler, J.Weed, P. Rigollet, (2017) `Near-linear time
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Advances in Neural Information Processing Systems (NIPS) 31

[23] Aude, G., Peyré, G., Cuturi, M., `Learning Generative Models with
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of the Twenty-First International Conference on Artficial Intelligence
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[24] Vayer, T., Chapel, L., Flamary, R., Tavenard, R. and Courty, N.
(2019). `Optimal Transport for structured data with application on
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of the 36th International Conference on Machine Learning (ICML).

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[26] Alaya M. Z., Bérar M., Gasso G., Rakotomamonjy A. (2019).
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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
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of the Twenty-Second International Conference on Artificial Intelligence
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[29] Chapel, L., Alaya, M., Gasso, G. (2020). `Partial Optimal Transport
with Applications on Positive-Unlabeled
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[31] Bonneel, Nicolas, et al. `Sliced and radon wasserstein barycenters
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[32] Huang, M., Ma S., Lai, L. (2021). `A Riemannian Block Coordinate
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[33] Kerdoncuff T., Emonet R., Marc S. `Sampled Gromov
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Machine Learning Journal (MJL), 2021

[34] Feydy, J., Séjourné, T., Vialard, F. X., Amari, S. I., Trouvé, A.,
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[35] Deshpande, I., Hu, Y. T., Sun, R., Pyrros, A., Siddiqui, N.,
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[36] Liutkus, A., Simsekli, U., Majewski, S., Durmus, A., & Stöter, F.
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[37] Janati, H., Cuturi, M., Gramfort, A. `Debiased sinkhorn
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[38] C. Vincent-Cuaz, T. Vayer, R. Flamary, M. Corneli, N. Courty,
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Conference on Machine Learning (ICML), 2021.

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