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Python Optimal Transport Library

Project description

POT: Python Optimal Transport

|PyPI version| |Build Status| |Documentation Status|

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 solver for the linear program/ Earth Movers Distance [1].
- Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2]
and stabilized version [9][10] with optional GPU implementation
(required cudamat).
- Bregman projections for Wasserstein barycenter [3] and unmixing [4].
- Optimal transport for domain adaptation with group lasso
regularization [5]
- Conditional gradient [6] and Generalized conditional gradient for
regularized OT [7].
- Joint OT matrix and mapping estimation [8].
- Wasserstein Discriminant Analysis [11] (requires autograd +
- Gromov-Wasserstein distances and barycenters [12]

Some demonstrations (both in Python and Jupyter Notebook format) are
available in the examples folder.


The library has been tested on Linux, MacOSX and Windows. It requires a
C++ compiler for using the EMD solver and relies on the following Python

- Numpy (>=1.11)
- Scipy (>=0.17)
- Cython (>=0.23)
- Matplotlib (>=1.5)

Pip installation

You can install the toolbox through PyPI with:


pip install POT

or get the very latest version by downloading it and then running:


python 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 <>`__. 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.


Some sub-modules require additional dependences which are discussed

- **ot.dr** (Wasserstein dimensionality rediuction) depends on autograd
and pymanopt that can be installed with:


pip install pymanopt autograd

- **ot.gpu** (GPU accelerated OT) depends on cudamat that have to be
installed with:


git clone
cd cudamat
python install --user # for user install (no root)

obviously you need CUDA installed and a compatible GPU.


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 is available on
`Readthedocs <>`__.

Here is a list of the Python notebooks available
`here <>`__ if you
want a quick look:

- `1D optimal
transport <>`__
- `OT Ground
Loss <>`__
- `Multiple EMD
computation <>`__
- `2D optimal transport on empirical
distributions <>`__
- `1D Wasserstein
barycenter <>`__
- `OT with user provided
regularization <>`__
- `Domain adaptation with optimal
transport <>`__
- `Color transfer in
images <>`__
- `OT mapping estimation for domain
adaptation <>`__
- `OT mapping estimation for color transfer in
images <>`__
- `Wasserstein Discriminant
Analysis <>`__
- `Gromov
Wasserstein <>`__
- `Gromov Wasserstein
Barycenter <>`__

You can also see the notebooks with `Jupyter
nbviewer <>`__.


The contributors to this library are:

- `Rémi Flamary <>`__
- `Nicolas Courty <>`__
- `Alexandre Gramfort <>`__
- `Laetitia Chapel <>`__
- `Michael Perrot <>`__
(Mapping estimation)
- `Léo Gautheron <>`__ (GPU implementation)
- `Nathalie
Gayraud <>`__
- `Stanislas Chambon <>`__
- `Antoine Rolet <>`__

This toolbox benefit a lot from open source research and we would like
to thank the following persons for providing some code (in various

- `Gabriel Peyré <>`__ (Wasserstein Barycenters
in Matlab)
- `Nicolas Bonneel <>`__ ( C++ code for
- `Marco Cuturi <>`__ (Sinkhorn Knopp in

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:

- On the `POT Slack channel <>`__
- On the POT `mailing
list <>`__

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


[1] Bonneel, N., Van De Panne, M., Paris, S., & Heidrich, W. (2011,
December). `Displacement interpolation using Lagrangian mass
transport <>`__.
In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p. 158). ACM.

[2] Cuturi, M. (2013). `Sinkhorn distances: Lightspeed computation of
optimal transport <>`__. 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 <>`__. 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 <>`__,
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 <>`__, 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 <>`__. 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 <>`__. arXiv preprint

[8] M. Perrot, N. Courty, R. Flamary, A. Habrard, `Mapping estimation
for discrete optimal
transport <>`__,
Neural Information Processing Systems (NIPS), 2016.

[9] Schmitzer, B. (2016). `Stabilized Sparse Scaling Algorithms for
Entropy Regularized Transport
Problems <>`__. arXiv preprint

[10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016).
`Scaling algorithms for unbalanced transport
problems <>`__. arXiv preprint

[11] Flamary, R., Cuturi, M., Courty, N., & Rakotomamonjy, A. (2016).
`Wasserstein Discriminant
Analysis <>`__. arXiv preprint

[12] Gabriel Peyré, Marco Cuturi, and Justin Solomon,
`Gromov-Wasserstein averaging of kernel and distance
matrices <>`__
International Conference on Machine Learning (ICML). 2016.

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