Metadata-Version: 2.4
Name: POT
Version: 0.9.6.post1
Summary: Python Optimal Transport Library
Home-page: https://github.com/PythonOT/POT
Download-URL: https://github.com/PythonOT/POT/archive/0.9.6.post1.tar.gz
Author: Remi Flamary, Nicolas Courty, Cédric Vincent-Cuaz, POT Contributors
Author-email: remi.flamary@gmail.com, ncourty@gmail.com, cedvincentcuaz@gmail.com
License: MIT
Platform: linux
Platform: macosx
Platform: windows
Classifier: Development Status :: 5 - Production/Stable
Classifier: Intended Audience :: Developers
Classifier: Intended Audience :: Education
Classifier: Intended Audience :: Science/Research
Classifier: License :: OSI Approved :: MIT License
Classifier: Environment :: Console
Classifier: Operating System :: OS Independent
Classifier: Operating System :: POSIX :: Linux
Classifier: Operating System :: MacOS
Classifier: Operating System :: POSIX
Classifier: Operating System :: Microsoft :: Windows
Classifier: Programming Language :: Python
Classifier: Programming Language :: C++
Classifier: Programming Language :: C
Classifier: Programming Language :: Cython
Classifier: Topic :: Utilities
Classifier: Topic :: Scientific/Engineering :: Artificial Intelligence
Classifier: Topic :: Scientific/Engineering :: Mathematics
Classifier: Topic :: Scientific/Engineering :: Information Analysis
Classifier: Programming Language :: Python :: 3
Classifier: Programming Language :: Python :: 3.7
Classifier: Programming Language :: Python :: 3.8
Classifier: Programming Language :: Python :: 3.9
Classifier: Programming Language :: Python :: 3.10
Classifier: Programming Language :: Python :: 3.11
Classifier: Programming Language :: Python :: 3.12
Requires-Python: >=3.7
Description-Content-Type: text/markdown
License-File: LICENSE
Requires-Dist: numpy>=1.16
Requires-Dist: scipy>=1.6
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Requires-Dist: matplotlib; extra == "all"
Dynamic: author
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# POT: Python Optimal Transport

[![PyPI version](https://badge.fury.io/py/POT.svg)](https://badge.fury.io/py/POT)
[![Anaconda Cloud](https://anaconda.org/conda-forge/pot/badges/version.svg)](https://anaconda.org/conda-forge/pot)
[![Build Status](https://github.com/PythonOT/POT/actions/workflows/build_tests.yml/badge.svg)](https://github.com/PythonOT/POT/actions)
[![Codecov Status](https://codecov.io/gh/PythonOT/POT/branch/master/graph/badge.svg)](https://codecov.io/gh/PythonOT/POT)
[![Downloads](https://static.pepy.tech/badge/pot)](https://pepy.tech/project/pot)
[![Anaconda downloads](https://anaconda.org/conda-forge/pot/badges/downloads.svg)](https://anaconda.org/conda-forge/pot)
[![License](https://anaconda.org/conda-forge/pot/badges/license.svg)](https://github.com/PythonOT/POT/blob/master/LICENSE)

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/](https://PythonOT.github.io/)

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


POT has the following main features:
* A large set of differentiable solvers for optimal transport problems, including:
  *  Exact linear OT, entropic and quadratic regularized OT,
  *  Gromov-Wasserstein (GW) distances, Fused GW distances and variants of
     quadratic OT,
  *  Unbalanced and partial OT for different divergences,
*  OT barycenters (Wasserstein and GW) for fixed and free support,
*  Fast OT solvers in 1D, on the circle and between Gaussian Mixture Models (GMMs),
*  Many ML related solvers, such as domain adaptation, optimal transport mapping
   estimation, subspace learning, Graph Neural Networks (GNNs) layers.
*  Several backends for easy use with Pytorch, Jax, Tensorflow, Numpy and Cupy arrays.

### Implemented Features

POT provides the following generic OT solvers:

* [OT Network Simplex solver](https://pythonot.github.io/auto_examples/plot_OT_1D.html) for the linear program/ Earth Movers Distance [1] .
* [Conditional gradient](https://pythonot.github.io/auto_examples/plot_optim_OTreg.html) [6] and [Generalized conditional gradient](https://pythonot.github.io/auto_examples/plot_optim_OTreg.html) for regularized OT [7].
* Entropic regularization OT solver with [Sinkhorn Knopp
  Algorithm](https://pythonot.github.io/auto_examples/plot_OT_1D.html) [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](https://pythonot.github.io/auto_examples/barycenters/plot_barycenter_lp_vs_entropic.html) [3], [convolutional barycenter](https://pythonot.github.io/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](https://pythonot.github.io/auto_examples/barycenters/plot_debiased_barycenter.html) [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] ](https://pythonot.github.io/auto_examples/barycenters/plot_barycenter_lp_vs_entropic.html) with LP solver (only small scale).
* [Gromov-Wasserstein distances](https://pythonot.github.io/auto_examples/gromov/plot_gromov.html) and [GW barycenters](https://pythonot.github.io/auto_examples/gromov/plot_gromov_barycenter.html)  (exact [13] and regularized [12,51]), differentiable using gradients from Graph Dictionary Learning [38]
 * [Fused-Gromov-Wasserstein distances solver](https://pythonot.github.io/auto_examples/gromov/plot_fgw.html#sphx-glr-auto-examples-plot-fgw-py) and [FGW barycenters](https://pythonot.github.io/auto_examples/gromov/plot_barycenter_fgw.html) (exact [24] and regularized [12,51]).
* [Stochastic
  solver](https://pythonot.github.io/auto_examples/others/plot_stochastic.html) and
  [differentiable losses](https://pythonot.github.io/auto_examples/backends/plot_stoch_continuous_ot_pytorch.html) for
  Large-scale Optimal Transport (semi-dual problem [18] and dual problem [19])
* [Sampled solver of Gromov Wasserstein](https://pythonot.github.io/auto_examples/gromov/plot_gromov.html) for large-scale problem with any loss functions [33]
* Non regularized [free support Wasserstein barycenters](https://pythonot.github.io/auto_examples/barycenters/plot_free_support_barycenter.html) [20].
* [One dimensional Unbalanced OT](https://pythonot.github.io/auto_examples/unbalanced-partial/plot_UOT_1D.html) with KL relaxation and [barycenter](https://pythonot.github.io/auto_examples/unbalanced-partial/plot_UOT_barycenter_1D.html) [10, 25]. Also [exact unbalanced OT](https://pythonot.github.io/auto_examples/unbalanced-partial/plot_unbalanced_ot.html) with KL and quadratic regularization and the [regularization path of UOT](https://pythonot.github.io/auto_examples/unbalanced-partial/plot_regpath.html) [41]
* [Partial Wasserstein and Gromov-Wasserstein](https://pythonot.github.io/auto_examples/unbalanced-partial/plot_partial_wass_and_gromov.html) and [Partial Fused Gromov-Wasserstein](https://pythonot.github.io/auto_examples/gromov/plot_partial_fgw.html) (exact [29] and entropic [3] formulations).
* [Sliced Wasserstein](https://pythonot.github.io/auto_examples/sliced-wasserstein/plot_variance.html) [31, 32] and Max-sliced Wasserstein [35] that can be used for gradient flows [36].
* [Wasserstein distance on the
  circle](https://pythonot.github.io/auto_examples/sliced-wasserstein/plot_compute_wasserstein_circle.html)
  [44, 45] and  [Spherical Sliced Wasserstein](https://pythonot.github.io/auto_examples/sliced-wasserstein/plot_variance_ssw.html) [46]
* [Graph Dictionary Learning solvers](https://pythonot.github.io/auto_examples/gromov/plot_gromov_wasserstein_dictionary_learning.html) [38].
* [Semi-relaxed (Fused) Gromov-Wasserstein divergences](https://pythonot.github.io/auto_examples/gromov/plot_semirelaxed_fgw.html) with corresponding [barycenter solvers](https://pythonot.github.io/auto_examples/gromov/plot_semirelaxed_gromov_wasserstein_barycenter.hmtl) (exact and regularized [48]).
* [Quantized (Fused) Gromov-Wasserstein distances](https://pythonot.github.io/auto_examples/gromov/plot_quantized_gromov_wasserstein.html) [68].
* [Efficient Discrete Multi Marginal Optimal Transport Regularization](https://pythonot.github.io/auto_examples/others/plot_demd_gradient_minimize.html) [50].
* [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/)/[Cupy](https://cupy.dev/)/[Tensorflow](https://www.tensorflow.org/) arrays.
* [Smooth Strongly Convex Nearest Brenier Potentials](https://pythonot.github.io/auto_examples/others/plot_SSNB.html#sphx-glr-auto-examples-others-plot-ssnb-py) [58], with an extension to bounding potentials using [59].
* [Gaussian Mixture Model OT](https://pythonot.github.io/auto_examples/gaussian_gmm/plot_GMMOT_plan.html#sphx-glr-auto-examples-others-plot-gmmot-plan-py) [69].
* [Co-Optimal Transport](https://pythonot.github.io/auto_examples/others/plot_COOT.html) [49] and
[unbalanced Co-Optimal Transport](https://pythonot.github.io/auto_examples/others/plot_learning_weights_with_COOT.html) [71].
* Fused unbalanced Gromov-Wasserstein [70].
* [Optimal Transport Barycenters for Generic Costs](https://pythonot.github.io/auto_examples/barycenters/plot_free_support_barycenter_generic_cost.html) [77]
* [Barycenters between Gaussian Mixture Models](https://pythonot.github.io/auto_examples/barycenters/plot_gmm_barycenter.html) [69, 77]

POT provides the following Machine Learning related solvers:

* [Optimal transport for domain
  adaptation](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_classes.html)
  with [group lasso regularization](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_classes.html),   [Laplacian regularization](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_laplacian.html) [5] [30] and [semi
  supervised setting](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_semi_supervised.html).
* [Linear OT mapping](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_linear_mapping.html) [14] and [Joint OT mapping estimation](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_mapping.html) [8].
* [Wasserstein Discriminant Analysis](https://pythonot.github.io/auto_examples/others/plot_WDA.html) [11] (requires autograd + pymanopt).
* [JCPOT algorithm for multi-source domain adaptation with target shift](https://pythonot.github.io/auto_examples/domain-adaptation/plot_otda_jcpot.html) [27].
* [Graph Neural Network OT layers TFGW](https://pythonot.github.io/auto_examples/gromov/plot_gnn_TFGW.html) [52] and TW (OT-GNN) [53]

Some other examples are available in the  [documentation](https://pythonot.github.io/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 references from the current version and from our [JMLR
paper](https://jmlr.org/papers/v22/20-451.html):

    Flamary R., Vincent-Cuaz C., Courty N., Gramfort A., Kachaiev O., Quang Tran H., David L., Bonet C., Cassereau N., Gnassounou T., Tanguy E., Delon J., Collas A., Mazelet S., Chapel L., Kerdoncuff T., Yu X., Feickert M., Krzakala P., Liu T., Fernandes Montesuma E. POT Python Optimal Transport (version 0.9.5). URL: https://github.com/PythonOT/POT

    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. URL: https://pythonot.github.io/

In Bibtex format:

```bibtex
@misc{flamary2024pot,
  author = {Flamary, R{\'e}mi and Vincent-Cuaz, C{\'e}dric and Courty, Nicolas and Gramfort, Alexandre and Kachaiev, Oleksii and Quang Tran, Huy and David, Laurène and Bonet, Cl{\'e}ment and Cassereau, Nathan and Gnassounou, Th{\'e}o and Tanguy, Eloi and Delon, Julie and Collas, Antoine and Mazelet, Sonia and Chapel, Laetitia and Kerdoncuff, Tanguy and Yu, Xizheng and Feickert, Matthew and Krzakala, Paul and Liu, Tianlin and Fernandes Montesuma, Eduardo},
  title = {POT Python Optimal Transport (version 0.9.5)},
  url = {https://github.com/PythonOT/POT},
  year = {2024}
}

@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:

```console
pip install POT
```

or get the very latest version by running:

```console
pip install -U https://github.com/PythonOT/POT/archive/master.zip # with --user for user install (no root)
```

Optional dependencies may be installed with
```console
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](https://conda-forge.org). To install it and the required dependencies:

```console
conda install -c conda-forge pot
```

#### Post installation check
After a correct installation, you should be able to import the module without errors:

```python
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:

```shell
pip install pymanopt autograd
```


## Examples

### Short examples

* Import the toolbox

```python
import ot
```

* Compute Wasserstein distances

```python
# a,b are 1D histograms (sum to 1 and positive)
# M is the ground cost matrix

# With the unified  API :
Wd = ot.solve(M, a, b).value # exact linear program
Wd_reg = ot.solve(M, a, b, reg=reg).value # entropic regularized OT

# With the old API :
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

```python
# a,b are 1D histograms (sum to 1 and positive)
# M is the ground cost matrix

# With the unified API :
T = ot.solve(M, a, b).plan # exact linear program
T_reg = ot.solve(M, a, b, reg=reg).plan # entropic regularized OT

# With the old API :
T = ot.emd(a, b, M) # exact linear program
T_reg = ot.sinkhorn(a, b, M, reg) # entropic regularized OT
```

* Compute OT on empirical distributions

```python
# X and Y are two 2D arrays of shape (n_samples, n_features)

# with squared euclidean metric
T = ot.solve_sample(X, Y).plan # exact linear program
T_reg = ot.solve_sample(X, Y, reg=reg).plan # entropic regularized OT

Wass_2 = ot.solve_sample(X, Y).value # Squared Wasserstein_2
Wass_1 = ot.solve_sample(X, Y, metric='euclidean').value # Wasserstein 1
```

* Compute Wasserstein barycenter

```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/](https://PythonOT.github.io/).

## Acknowledgements

This toolbox has been created by [Rémi Flamary](https://remi.flamary.com/) and [Nicolas Courty](http://people.irisa.fr/Nicolas.Courty/).

It is currently maintained by :

* [Rémi Flamary](https://remi.flamary.com/)
* [Cédric Vincent-Cuaz](https://cedricvincentcuaz.github.io/)

The POT contributors to this library are listed [here](CONTRIBUTORS.md).

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

ANRCNRS3IAHi!PARIS

## Contributions and code of conduct

Every contribution is welcome and should respect the [contribution guidelines](https://pythonot.github.io/master/contributing.html). Each member of the project is expected to follow the [code of conduct](https://pythonot.github.io/master/code_of_conduct.html).

## 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), Workshop 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

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