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POT: Python Optimal Transport

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This open source Python library provides several solvers for optimization problems related to Optimal Transport for signal, image processing and machine learning.

Website and documentation:

Source Code (MIT):

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

POT provides the following Machine Learning related solvers:

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.

In Bibtex format:

  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     = {}


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:

pip install POT

or get the very latest version by running:

pip install -U # with --user for user install (no root)

Optional dependencies may be installed with

pip install POT[all]

Note that this installs cvxopt, which is licensed under GPL 3.0. Alternatively, if you cannot use GPL-licensed software, the specific optional dependencies may be installed individually, or per-submodule. The available optional installations are backend-jax, backend-tf, backend-torch, cvxopt, dr, gnn, all.

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 named ot instead of pot.


Some sub-modules require additional dependencies which are discussed below

  • ot.dr (Wasserstein dimensionality reduction) depends on autograd and pymanopt that can be installed with:
pip install pymanopt autograd


Short 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

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


This toolbox has been created by

It is currently maintained by

The numerous contributors to this library are listed here.

POT has benefited from the financing or manpower from the following partners:


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.


You can ask questions and join the development discussion:

You can also post bug reports and feature requests in Github issues. Make sure to read our guidelines first.


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