# POT: Python Optimal Transport

## Contents

### POT: Python Optimal Transport

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: 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 for the linear program/ Earth Movers Distance [1] .

Conditional gradient [6] and Generalized conditional gradient for regularized OT [7].

Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2] , stabilized version [9] [10] [34], lazy CPU/GPU solver from geomloss [60] [61], greedy Sinkhorn [22] and Screening Sinkhorn [26] .

Bregman projections for Wasserstein barycenter [3], convolutional barycenter [21] and unmixing [4].

Sinkhorn divergence [23] and entropic regularization OT from empirical data.

Debiased Sinkhorn barycenters Sinkhorn divergence barycenter [37]

Smooth optimal transport solvers (dual and semi-dual) for KL and squared L2 regularizations [17].

Weak OT solver between empirical distributions [39]

Non regularized Wasserstein barycenters [16] with LP solver (only small scale).

Gromov-Wasserstein distances and GW barycenters (exact [13] and regularized [12,51]), differentiable using gradients from Graph Dictionary Learning [38]

Fused-Gromov-Wasserstein distances solver and FGW barycenters (exact [24] and regularized [12,51]).

Stochastic solver and differentiable losses for Large-scale Optimal Transport (semi-dual problem [18] and dual problem [19])

Sampled solver of Gromov Wasserstein for large-scale problem with any loss functions [33]

Non regularized free support Wasserstein barycenters [20].

One dimensional Unbalanced OT with KL relaxation and barycenter [10, 25]. Also exact unbalanced OT with KL and quadratic regularization and the regularization path of UOT [41]

Partial Wasserstein and Gromov-Wasserstein (exact [29] and entropic [3] formulations).

Sliced Wasserstein [31, 32] and Max-sliced Wasserstein [35] that can be used for gradient flows [36].

Wasserstein distance on the circle [44, 45]

Semi-relaxed (Fused) Gromov-Wasserstein divergences with corresponding barycenter solvers (exact and regularized [48]).

Efficient Discrete Multi Marginal Optimal Transport Regularization [50].

Several backends for easy use of POT with Pytorch/jax/Numpy/Cupy/Tensorflow arrays.

Smooth Strongly Convex Nearest Brenier Potentials [58], with an extension to bounding potentials using [59].

Gaussian Mixture Model OT [69]

POT provides the following Machine Learning related solvers:

Optimal transport for domain adaptation with group lasso regularization, Laplacian regularization [5] [30] and semi supervised setting.

Linear OT mapping [14] and Joint OT mapping estimation [8].

Wasserstein Discriminant Analysis [11] (requires autograd + pymanopt).

JCPOT algorithm for multi-source domain adaptation with target shift [27].

Graph Neural Network OT layers TFGW [52] and TW (OT-GNN) [53]

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.
Website: https://pythonot.github.io/
```

In Bibtex format:

```
@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:

```
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)
```

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`

.

##### Dependencies

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
```

#### Examples

##### 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 https://PythonOT.github.io/.

#### Acknowledgements

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.

#### Support

You can ask questions and join the development discussion:

On the POT slack channel

On the POT gitter 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.

#### References

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