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<!DOCTYPE html>
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<sectionid="pot-python-optimal-transport">
<h1>POT: Python Optimal Transport<aclass="headerlink" href="#pot-python-optimal-transport" title="Permalink to this headline"></a></h1>
<p><aclass="reference external" href="https://badge.fury.io/py/POT"><imgalt="PyPI version" src="https://badge.fury.io/py/POT.svg" /></a><aclass="reference external" href="https://anaconda.org/conda-forge/pot"><imgalt="Anaconda Cloud" src="https://anaconda.org/conda-forge/pot/badges/version.svg" /></a><aclass="reference external" href="https://github.com/PythonOT/POT/actions"><imgalt="Build Status" src="https://github.com/PythonOT/POT/workflows/build/badge.svg?branch=master&event=push" /></a><aclass="reference external" href="https://codecov.io/gh/PythonOT/POT"><imgalt="Codecov Status" src="https://codecov.io/gh/PythonOT/POT/branch/master/graph/badge.svg" /></a>
<aclass="reference external" href="https://pepy.tech/project/pot"><imgalt="Downloads" src="https://pepy.tech/badge/pot" /></a><aclass="reference external" href="https://anaconda.org/conda-forge/pot"><imgalt="Anaconda downloads" src="https://anaconda.org/conda-forge/pot/badges/downloads.svg" /></a><aclass="reference external" href="https://github.com/PythonOT/POT/blob/master/LICENSE"><imgalt="License" src="https://anaconda.org/conda-forge/pot/badges/license.svg" /></a></p>
<p>This open source Python library provide several solvers for optimization
problems related to Optimal Transport for signal, image processing and
machine learning.</p>
<p>Website and documentation: <aclass="reference external" href="https://PythonOT.github.io/">https://PythonOT.github.io/</a></p>
<p>Source Code (MIT): <aclass="reference external" href="https://github.com/PythonOT/POT">https://github.com/PythonOT/POT</a></p>
<p>POT provides the following generic OT solvers (links to examples):</p>
<ulclass="simple">
<li><p><aclass="reference external" href="auto_examples/plot_OT_1D.html">OT Network Simplex
solver</a>
for the linear program/ Earth Movers Distance [1] .</p></li>
<li><p><aclass="reference external" href="auto_examples/plot_optim_OTreg.html">Conditional
gradient</a>
[6] and <aclass="reference external" href="auto_examples/plot_optim_OTreg.html">Generalized conditional
gradient</a>
for regularized OT [7].</p></li>
<li><p>Entropic regularization OT solver with <aclass="reference external" href="auto_examples/plot_OT_1D.html">Sinkhorn Knopp
Algorithm</a>
[2] , stabilized version [9] [10] [34], greedy Sinkhorn [22] and
<aclass="reference external" href="auto_examples/plot_screenkhorn_1D.html">Screening Sinkhorn
[26]</a>.</p></li>
<li><p>Bregman projections for <aclass="reference external" href="auto_examples/barycenters/plot_barycenter_lp_vs_entropic.html">Wasserstein
barycenter</a>
[3], <aclass="reference external" href="auto_examples/barycenters/plot_convolutional_barycenter.html">convolutional
barycenter</a>
[21] and unmixing [4].</p></li>
<li><p>Sinkhorn divergence [23] and entropic regularization OT from
empirical data.</p></li>
<li><p>Debiased Sinkhorn barycenters <aclass="reference external" href="auto_examples/barycenters/plot_debiased_barycenter.html">Sinkhorn divergence
barycenter</a>
[37]</p></li>
<li><p><aclass="reference external" href="auto_examples/plot_OT_1D_smooth.html">Smooth optimal transport
solvers</a>
(dual and semi-dual) for KL and squared L2 regularizations [17].</p></li>
<li><p>Non regularized <aclass="reference external" href="auto_examples/barycenters/plot_barycenter_lp_vs_entropic.html">Wasserstein barycenters
[16]</a>)
with LP solver (only small scale).</p></li>
<li><p><aclass="reference external" href="auto_examples/gromov/plot_gromov.html">Gromov-Wasserstein
distances</a>
and <aclass="reference external" href="auto_examples/gromov/plot_gromov_barycenter.html">GW
barycenters</a>
(exact [13] and regularized [12]), differentiable using gradients
from</p></li>
<li><p><aclass="reference external" href="auto_examples/gromov/plot_fgw.html#sphx-glr-auto-examples-plot-fgw-py">Fused-Gromov-Wasserstein distances
solver</a>
and <aclass="reference external" href="auto_examples/gromov/plot_barycenter_fgw.html">FGW
barycenters</a>
[24]</p></li>
<li><p><aclass="reference external" href="auto_examples/plot_stochastic.html">Stochastic
solver</a>
for Large-scale Optimal Transport (semi-dual problem [18] and dual
problem [19])</p></li>
<li><p><aclass="reference external" href="auto_examples/gromov/plot_gromov.html">Stochastic solver of Gromov
Wasserstein</a>
for large-scale problem with any loss functions [33]</p></li>
<li><p>Non regularized <aclass="reference external" href="auto_examples/barycenters/plot_free_support_barycenter.html">free support Wasserstein
barycenters</a>
[20].</p></li>
<li><p><aclass="reference external" href="auto_examples/unbalanced-partial/plot_UOT_1D.html">Unbalanced
OT</a>
with KL relaxation and
<aclass="reference external" href="auto_examples/unbalanced-partial/plot_UOT_barycenter_1D.html">barycenter</a>
[10, 25].</p></li>
<li><p><aclass="reference external" href="auto_examples/unbalanced-partial/plot_partial_wass_and_gromov.html">Partial Wasserstein and
Gromov-Wasserstein</a>
(exact [29] and entropic [3] formulations).</p></li>
<li><p><aclass="reference external" href="auto_examples/sliced-wasserstein/plot_variance.html">Sliced
Wasserstein</a>
[31, 32] and Max-sliced Wasserstein [35] that can be used for
gradient flows [36].</p></li>
<li><p><aclass="reference external" href="https://pythonot.github.io/quickstart.html#solving-ot-with-multiple-backends">Several
backends</a>
for easy use of POT with
<aclass="reference external" href="https://pytorch.org/">Pytorch</a>/<aclass="reference external" href="https://github.com/google/jax">jax</a>/<aclass="reference external" href="https://numpy.org/">Numpy</a>
arrays.</p></li>
</ul>
<p>POT provides the following Machine Learning related solvers:</p>
<ulclass="simple">
<li><p><aclass="reference external" href="auto_examples/domain-adaptation/plot_otda_classes.html">Optimal transport for domain
adaptation</a>
with <aclass="reference external" href="auto_examples/domain-adaptation/plot_otda_classes.html">group lasso
regularization</a>,
<aclass="reference external" href="auto_examples/domain-adaptation/plot_otda_laplacian.html">Laplacian
regularization</a>
[5] [30] and <aclass="reference external" href="auto_examples/domain-adaptation/plot_otda_semi_supervised.html">semi supervised
setting</a>.</p></li>
<li><p><aclass="reference external" href="auto_examples/domain-adaptation/plot_otda_linear_mapping.html">Linear OT
mapping</a>
[14] and <aclass="reference external" href="auto_examples/domain-adaptation/plot_otda_mapping.html">Joint OT mapping
estimation</a>
[8].</p></li>
<li><p><aclass="reference external" href="auto_examples/others/plot_WDA.html">Wasserstein Discriminant
Analysis</a>
[11] (requires autograd + pymanopt).</p></li>
<li><p><aclass="reference external" href="auto_examples/domain-adaptation/plot_otda_jcpot.html">JCPOT algorithm for multi-source domain adaptation with target
shift</a>
[27].</p></li>
</ul>
<p>Some other examples are available in the
<aclass="reference external" href="auto_examples/index.html">documentation</a>.</p>
<sectionid="using-and-citing-the-toolbox">
<h2>Using and citing the toolbox<aclass="headerlink" href="#using-and-citing-the-toolbox" title="Permalink to this headline"></a></h2>
<p>If you use this toolbox in your research and find it useful, please cite
POT using the following reference from our <aclass="reference external" href="https://jmlr.org/papers/v22/20-451.html">JMLR
paper</a>:</p>
<divclass="highlight-default notranslate"><divclass="highlight"><pre><span></span>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/
</pre></div>
</div>
<p>In Bibtex format:</p>
<divclass="highlight-bibtex notranslate"><divclass="highlight"><pre><span></span><spanclass="nc">@article</span><spanclass="p">{</span><spanclass="nl">flamary2021pot</span><spanclass="p">,</span>
<spanclass="na">author</span><spanclass="p">=</span><spanclass="s">{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}</span><spanclass="p">,</span>
<spanclass="na">title</span><spanclass="p">=</span><spanclass="s">{POT: Python Optimal Transport}</span><spanclass="p">,</span>
<spanclass="na">journal</span><spanclass="p">=</span><spanclass="s">{Journal of Machine Learning Research}</span><spanclass="p">,</span>
<spanclass="na">year</span><spanclass="p">=</span><spanclass="s">{2021}</span><spanclass="p">,</span>
<spanclass="na">volume</span><spanclass="p">=</span><spanclass="s">{22}</span><spanclass="p">,</span>
<spanclass="na">number</span><spanclass="p">=</span><spanclass="s">{78}</span><spanclass="p">,</span>
<spanclass="na">pages</span><spanclass="p">=</span><spanclass="s">{1-8}</span><spanclass="p">,</span>
<spanclass="na">url</span><spanclass="p">=</span><spanclass="s">{http://jmlr.org/papers/v22/20-451.html}</span>
<spanclass="p">}</span>
</pre></div>
</div>
<sectionid="installation">
<h3>Installation<aclass="headerlink" href="#installation" title="Permalink to this headline"></a></h3>
<p>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:</p>
<ulclass="simple">
<li><p>Numpy (>=1.16)</p></li>
<li><p>Scipy (>=1.0)</p></li>
<li><p>Cython (>=0.23) (build only, not necessary when installing from pip
or conda)</p></li>
</ul>
</section>
</section>
<sectionid="pip-installation">
<h2>Pip installation<aclass="headerlink" href="#pip-installation" title="Permalink to this headline"></a></h2>
<p>You can install the toolbox through PyPI with:</p>
<divclass="highlight-console notranslate"><divclass="highlight"><pre><span></span><spanclass="go">pip install POT</span>
</pre></div>
</div>
<p>or get the very latest version by running:</p>
<divclass="highlight-console notranslate"><divclass="highlight"><pre><span></span><spanclass="go">pip install -U https://github.com/PythonOT/POT/archive/master.zip # with --user for user install (no root)</span>
</pre></div>
</div>
</section>
<sectionid="anaconda-installation-with-conda-forge">
<h2>Anaconda installation with conda-forge<aclass="headerlink" href="#anaconda-installation-with-conda-forge" title="Permalink to this headline"></a></h2>
<p>If you use the Anaconda python distribution, POT is available in
<aclass="reference external" href="https://conda-forge.org">conda-forge</a>. To install it and the
required dependencies:</p>
<divclass="highlight-console notranslate"><divclass="highlight"><pre><span></span><spanclass="go">conda install -c conda-forge pot</span>
</pre></div>
</div>
</section>
<sectionid="post-installation-check">
<h2>Post installation check<aclass="headerlink" href="#post-installation-check" title="Permalink to this headline"></a></h2>
<p>After a correct installation, you should be able to import the module
without errors:</p>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="kn">import</span><spanclass="nn">ot</span>
</pre></div>
</div>
<p>Note that for easier access the module is named <codeclass="docutils literal notranslate"><spanclass="pre">ot</span></code> instead of
<codeclass="docutils literal notranslate"><spanclass="pre">pot</span></code>.</p>
<p>Some sub-modules require additional dependences which are discussed
below</p>
<ulclass="simple">
<li><p><strong>ot.dr</strong> (Wasserstein dimensionality reduction) depends on autograd
and pymanopt that can be installed with:</p></li>
</ul>
<divclass="highlight-shell notranslate"><divclass="highlight"><pre><span></span>pip install pymanopt autograd
</pre></div>
</div>
<ulclass="simple">
<li><p><strong>ot.gpu</strong> (GPU accelerated OT) depends on cupy that have to be
installed following instructions on <aclass="reference external" href="https://docs-cupy.chainer.org/en/stable/install.html">this
page</a>.
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.</p></li>
</ul>
<sectionid="examples">
<h3>Examples<aclass="headerlink" href="#examples" title="Permalink to this headline"></a></h3>
<sectionid="short-examples">
<h4>Short examples<aclass="headerlink" href="#short-examples" title="Permalink to this headline"></a></h4>
<ulclass="simple">
<li><p>Import the toolbox</p></li>
</ul>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="kn">import</span><spanclass="nn">ot</span>
</pre></div>
</div>
<ulclass="simple">
<li><p>Compute Wasserstein distances</p></li>
</ul>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># a,b are 1D histograms (sum to 1 and positive)</span>
<spanclass="c1"># M is the ground cost matrix</span>
<spanclass="n">Wd</span><spanclass="o">=</span><spanclass="n">ot</span><spanclass="o">.</span><spanclass="n">emd2</span><spanclass="p">(</span><spanclass="n">a</span><spanclass="p">,</span><spanclass="n">b</span><spanclass="p">,</span><spanclass="n">M</span><spanclass="p">)</span><spanclass="c1"># exact linear program</span>
<spanclass="n">Wd_reg</span><spanclass="o">=</span><spanclass="n">ot</span><spanclass="o">.</span><spanclass="n">sinkhorn2</span><spanclass="p">(</span><spanclass="n">a</span><spanclass="p">,</span><spanclass="n">b</span><spanclass="p">,</span><spanclass="n">M</span><spanclass="p">,</span><spanclass="n">reg</span><spanclass="p">)</span><spanclass="c1"># entropic regularized OT</span>
<spanclass="c1"># if b is a matrix compute all distances to a and return a vector</span>
</pre></div>
</div>
<ulclass="simple">
<li><p>Compute OT matrix</p></li>
</ul>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># a,b are 1D histograms (sum to 1 and positive)</span>
<spanclass="c1"># M is the ground cost matrix</span>
<spanclass="n">T</span><spanclass="o">=</span><spanclass="n">ot</span><spanclass="o">.</span><spanclass="n">emd</span><spanclass="p">(</span><spanclass="n">a</span><spanclass="p">,</span><spanclass="n">b</span><spanclass="p">,</span><spanclass="n">M</span><spanclass="p">)</span><spanclass="c1"># exact linear program</span>
<spanclass="n">T_reg</span><spanclass="o">=</span><spanclass="n">ot</span><spanclass="o">.</span><spanclass="n">sinkhorn</span><spanclass="p">(</span><spanclass="n">a</span><spanclass="p">,</span><spanclass="n">b</span><spanclass="p">,</span><spanclass="n">M</span><spanclass="p">,</span><spanclass="n">reg</span><spanclass="p">)</span><spanclass="c1"># entropic regularized OT</span>
</pre></div>
</div>
<ulclass="simple">
<li><p>Compute Wasserstein barycenter</p></li>
</ul>
<divclass="highlight-python notranslate"><divclass="highlight"><pre><span></span><spanclass="c1"># A is a n*d matrix containing d 1D histograms</span>
<spanclass="c1"># M is the ground cost matrix</span>
<spanclass="n">ba</span><spanclass="o">=</span><spanclass="n">ot</span><spanclass="o">.</span><spanclass="n">barycenter</span><spanclass="p">(</span><spanclass="n">A</span><spanclass="p">,</span><spanclass="n">M</span><spanclass="p">,</span><spanclass="n">reg</span><spanclass="p">)</span><spanclass="c1"># reg is regularization parameter</span>
</pre></div>
</div>
</section>
<sectionid="examples-and-notebooks">
<h4>Examples and Notebooks<aclass="headerlink" href="#examples-and-notebooks" title="Permalink to this headline"></a></h4>
<p>The examples folder contain several examples and use case for the
library. The full documentation with examples and output is available on
<aclass="reference external" href="https://PythonOT.github.io/">https://PythonOT.github.io/</a>.</p>
</section>
</section>
<sectionid="acknowledgements">
<h3>Acknowledgements<aclass="headerlink" href="#acknowledgements" title="Permalink to this headline"></a></h3>
<p>This toolbox has been created and is maintained by</p>
<ulclass="simple">
<li><p><aclass="reference external" href="http://remi.flamary.com/">Rémi Flamary</a></p></li>
<li><p><aclass="reference external" href="http://people.irisa.fr/Nicolas.Courty/">Nicolas Courty</a></p></li>
</ul>
<p>The contributors to this library are</p>
<ulclass="simple">
<li><p><aclass="reference external" href="http://alexandre.gramfort.net/">Alexandre Gramfort</a> (CI,
documentation)</p></li>
<li><p><aclass="reference external" href="http://people.irisa.fr/Laetitia.Chapel/">Laetitia Chapel</a>
(Partial OT)</p></li>
<li><p><aclass="reference external" href="http://perso.univ-st-etienne.fr/pem82055/">Michael Perrot</a>
(Mapping estimation)</p></li>
<li><p><aclass="reference external" href="https://github.com/aje">Léo Gautheron</a> (GPU implementation)</p></li>
<li><p><aclass="reference external" href="https://www.linkedin.com/in/nathalie-t-h-gayraud/?ppe=1">Nathalie
Gayraud</a>
(DA classes)</p></li>
<li><p><aclass="reference external" href="https://slasnista.github.io/">Stanislas Chambon</a> (DA classes)</p></li>
<li><p><aclass="reference external" href="https://arolet.github.io/">Antoine Rolet</a> (EMD solver debug)</p></li>
<li><p>Erwan Vautier (Gromov-Wasserstein)</p></li>
<li><p><aclass="reference external" href="https://kilianfatras.github.io/">Kilian Fatras</a> (Stochastic
solvers)</p></li>
<li><p><aclass="reference external" href="https://sites.google.com/site/alainrakotomamonjy/home">Alain
Rakotomamonjy</a></p></li>
<li><p><aclass="reference external" href="https://tvayer.github.io/">Vayer Titouan</a> (Gromov-Wasserstein -,
Fused-Gromov-Wasserstein)</p></li>
<li><p><aclass="reference external" href="https://hichamjanati.github.io/">Hicham Janati</a> (Unbalanced OT,
Debiased barycenters)</p></li>
<li><p><aclass="reference external" href="https://rtavenar.github.io/">Romain Tavenard</a> (1d Wasserstein)</p></li>
<li><p><aclass="reference external" href="http://mzalaya.github.io/">Mokhtar Z. Alaya</a> (Screenkhorn)</p></li>
<li><p><aclass="reference external" href="https://ievred.github.io/">Ievgen Redko</a> (Laplacian DA, JCPOT)</p></li>
<li><p><aclass="reference external" href="https://adriencorenflos.github.io/">Adrien Corenflos</a> (Sliced
Wasserstein Distance)</p></li>
<li><p><aclass="reference external" href="https://hv0nnus.github.io/">Tanguy Kerdoncuff</a> (Sampled Gromov
Wasserstein)</p></li>
<li><p><aclass="reference external" href="https://mhhuang95.github.io">Minhui Huang</a> (Projection Robust
Wasserstein Distance)</p></li>
</ul>
<p>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):</p>
<ulclass="simple">
<li><p><aclass="reference external" href="http://gpeyre.github.io/">Gabriel Peyré</a> (Wasserstein Barycenters
in Matlab)</p></li>
<li><p><aclass="reference external" href="https://mblondel.org/">Mathieu Blondel</a> (original implementation
smooth OT)</p></li>
<li><p><aclass="reference external" href="http://liris.cnrs.fr/~nbonneel/">Nicolas Bonneel</a> ( C++ code for
EMD)</p></li>
<li><p><aclass="reference external" href="http://marcocuturi.net/">Marco Cuturi</a> (Sinkhorn Knopp in
Matlab/Cuda)</p></li>
</ul>
</section>
<sectionid="contributions-and-code-of-conduct">
<h3>Contributions and code of conduct<aclass="headerlink" href="#contributions-and-code-of-conduct" title="Permalink to this headline"></a></h3>
<p>Every contribution is welcome and should respect the <aclass="reference external" href=".github/CONTRIBUTING.md">contribution
guidelines</a>. Each member of the project is
expected to follow the <aclass="reference external" href=".github/CODE_OF_CONDUCT.md">code of conduct</a>.</p>
</section>
<sectionid="support">
<h3>Support<aclass="headerlink" href="#support" title="Permalink to this headline"></a></h3>
<p>You can ask questions and join the development discussion:</p>
<ulclass="simple">
<li><p>On the POT <aclass="reference external" href="https://pot-toolbox.slack.com">slack channel</a></p></li>
<li><p>On the POT <aclass="reference external" href="https://gitter.im/PythonOT/community">gitter channel</a></p></li>
<li><p>On the POT <aclass="reference external" href="https://mail.python.org/mm3/mailman3/lists/pot.python.org/">mailing
list</a></p></li>
</ul>
<p>You can also post bug reports and feature requests in Github issues.
Make sure to read our <aclass="reference external" href=".github/CONTRIBUTING.md">guidelines</a> first.</p>
</section>
<sectionid="references">
<h3>References<aclass="headerlink" href="#references" title="Permalink to this headline"></a></h3>
<p>[1] Bonneel, N., Van De Panne, M., Paris, S., & Heidrich, W. (2011,
December). <aclass="reference external" href="https://people.csail.mit.edu/sparis/publi/2011/sigasia/Bonneel_11_Displacement_Interpolation.pdf">Displacement interpolation using Lagrangian mass
transport</a>.
In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p. 158). ACM.</p>
<p>[2] Cuturi, M. (2013). <aclass="reference external" href="https://arxiv.org/pdf/1306.0895.pdf">Sinkhorn distances: Lightspeed computation of
optimal transport</a>. In Advances
in Neural Information Processing Systems (pp. 2292-2300).</p>
<p>[3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G.
(2015). <aclass="reference external" href="https://arxiv.org/pdf/1412.5154.pdf">Iterative Bregman projections for regularized transportation
problems</a>. SIAM Journal on
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<p>[4] S. Nakhostin, N. Courty, R. Flamary, D. Tuia, T. Corpetti,
<aclass="reference external" href="https://hal.archives-ouvertes.fr/hal-01377236/document">Supervised planetary unmixing with optimal
transport</a>,
Whorkshop on Hyperspectral Image and Signal Processing : Evolution in
Remote Sensing (WHISPERS), 2016.</p>
<p>[5] N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, <aclass="reference external" href="https://arxiv.org/pdf/1507.00504.pdf">Optimal Transport
for Domain Adaptation</a>, in IEEE
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<p>[6] Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014).
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<p>[7] Rakotomamonjy, A., Flamary, R., & Courty, N. (2015). <aclass="reference external" href="https://arxiv.org/pdf/1510.06567.pdf">Generalized
conditional gradient: analysis of convergence and
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arXiv:1510.06567.</p>
<p>[8] M. Perrot, N. Courty, R. Flamary, A. Habrard (2016), <aclass="reference external" href="http://remi.flamary.com/biblio/perrot2016mapping.pdf">Mapping
estimation for discrete optimal
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<p>[9] Schmitzer, B. (2016). <aclass="reference external" href="https://arxiv.org/pdf/1610.06519.pdf">Stabilized Sparse Scaling Algorithms for
Entropy Regularized Transport
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<p>[10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016).
<aclass="reference external" href="https://arxiv.org/pdf/1607.05816.pdf">Scaling algorithms for unbalanced transport
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<p>[11] Flamary, R., Cuturi, M., Courty, N., & Rakotomamonjy, A. (2016).
<aclass="reference external" href="https://arxiv.org/pdf/1608.08063.pdf">Wasserstein Discriminant
Analysis</a>. arXiv preprint
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<p>[12] Gabriel Peyré, Marco Cuturi, and Justin Solomon (2016),
<aclass="reference external" href="http://proceedings.mlr.press/v48/peyre16.html">Gromov-Wasserstein averaging of kernel and distance
matrices</a>
International Conference on Machine Learning (ICML).</p>
<p>[13] Mémoli, Facundo (2011). <aclass="reference external" href="https://media.adelaide.edu.au/acvt/Publications/2011/2011-Gromov%E2%80%93Wasserstein%20Distances%20and%20the%20Metric%20Approach%20to%20Object%20Matching.pdf">Gromov–Wasserstein distances and the
metric approach to object
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Foundations of computational mathematics 11.4 : 417-487.</p>
<p>[14] Knott, M. and Smith, C. S. (1984).`On the optimal mapping of
distributions <<aclass="reference external" href="https://link.springer.com/article/10.1007/BF00934745">https://link.springer.com/article/10.1007/BF00934745</a>>`__,
Journal of Optimization Theory and Applications Vol 43.</p>
<p>[15] Peyré, G., & Cuturi, M. (2018). <aclass="reference external" href="https://arxiv.org/pdf/1803.00567.pdf">Computational Optimal
Transport</a> .</p>
<p>[16] Agueh, M., & Carlier, G. (2011). <aclass="reference external" href="https://hal.archives-ouvertes.fr/hal-00637399/document">Barycenters in the Wasserstein
space</a>. SIAM
Journal on Mathematical Analysis, 43(2), 904-924.</p>
<p>[17] Blondel, M., Seguy, V., & Rolet, A. (2018). <aclass="reference external" href="https://arxiv.org/abs/1710.06276">Smooth and Sparse
Optimal Transport</a>. Proceedings of
the Twenty-First International Conference on Artificial Intelligence and
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<p>[18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016) <aclass="reference external" href="https://arxiv.org/abs/1605.08527">Stochastic
Optimization for Large-scale Optimal
Transport</a>. Advances in Neural
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<p>[19] Seguy, V., Bhushan Damodaran, B., Flamary, R., Courty, N., Rolet,
A.& Blondel, M. <aclass="reference external" href="https://arxiv.org/pdf/1711.02283.pdf">Large-scale Optimal Transport and Mapping
Estimation</a>. International
Conference on Learning Representation (2018)</p>
<p>[20] Cuturi, M. and Doucet, A. (2014) <aclass="reference external" href="http://proceedings.mlr.press/v32/cuturi14.html">Fast Computation of Wasserstein
Barycenters</a>.
International Conference in Machine Learning</p>
<p>[21] Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher, A.,
Nguyen, A. & Guibas, L. (2015). <aclass="reference external" href="https://dl.acm.org/citation.cfm?id=2766963">Convolutional wasserstein distances:
Efficient optimal transportation on geometric
domains</a>. ACM
Transactions on Graphics (TOG), 34(4), 66.</p>
<p>[22] J. Altschuler, J.Weed, P. Rigollet, (2017) <aclass="reference external" href="https://papers.nips.cc/paper/6792-near-linear-time-approximation-algorithms-for-optimal-transport-via-sinkhorn-iteration.pdf">Near-linear time
approximation algorithms for optimal transport via Sinkhorn
iteration</a>,
Advances in Neural Information Processing Systems (NIPS) 31</p>
<p>[23] Aude, G., Peyré, G., Cuturi, M., <aclass="reference external" href="https://arxiv.org/abs/1706.00292">Learning Generative Models with
Sinkhorn Divergences</a>, Proceedings
of the Twenty-First International Conference on Artficial Intelligence
and Statistics, (AISTATS) 21, 2018</p>
<p>[24] Vayer, T., Chapel, L., Flamary, R., Tavenard, R. and Courty, N.
(2019). <aclass="reference external" href="http://proceedings.mlr.press/v97/titouan19a.html">Optimal Transport for structured data with application on
graphs</a> Proceedings
of the 36th International Conference on Machine Learning (ICML).</p>
<p>[25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. (2015).
<aclass="reference external" href="http://cbcl.mit.edu/wasserstein/">Learning with a Wasserstein Loss</a>
Advances in Neural Information Processing Systems (NIPS).</p>
<p>[26] Alaya M. Z., Bérar M., Gasso G., Rakotomamonjy A. (2019).
<aclass="reference external" href="https://papers.nips.cc/paper/9386-screening-sinkhorn-algorithm-for-regularized-optimal-transport">Screening Sinkhorn Algorithm for Regularized Optimal
Transport</a>,
Advances in Neural Information Processing Systems 33 (NeurIPS).</p>
<p>[27] Redko I., Courty N., Flamary R., Tuia D. (2019). <aclass="reference external" href="http://proceedings.mlr.press/v89/redko19a.html">Optimal Transport
for Multi-source Domain Adaptation under Target
Shift</a>, Proceedings
of the Twenty-Second International Conference on Artificial Intelligence
and Statistics (AISTATS) 22, 2019.</p>
<p>[28] Caffarelli, L. A., McCann, R. J. (2010). <aclass="reference external" href="http://www.math.toronto.edu/~mccann/papers/annals2010.pdf">Free boundaries in
optimal transport and Monge-Ampere obstacle
problems</a>,
Annals of mathematics, 673-730.</p>
<p>[29] Chapel, L., Alaya, M., Gasso, G. (2020). <aclass="reference external" href="https://arxiv.org/abs/2002.08276">Partial Optimal Transport
with Applications on Positive-Unlabeled
Learning</a>, Advances in Neural
Information Processing Systems (NeurIPS), 2020.</p>
<p>[30] Flamary R., Courty N., Tuia D., Rakotomamonjy A. (2014). <aclass="reference external" href="https://remi.flamary.com/biblio/flamary2014optlaplace.pdf">Optimal
transport with Laplacian regularization: Applications to domain
adaptation and shape
matching</a>,
NIPS Workshop on Optimal Transport and Machine Learning OTML, 2014.</p>
<p>[31] Bonneel, Nicolas, et al. <aclass="reference external" href="https://perso.liris.cnrs.fr/nicolas.bonneel/WassersteinSliced-JMIV.pdf">Sliced and radon wasserstein barycenters
of
measures</a>,
Journal of Mathematical Imaging and Vision 51.1 (2015): 22-45</p>
<p>[32] Huang, M., Ma S., Lai, L. (2021). <aclass="reference external" href="http://proceedings.mlr.press/v139/huang21e.html">A Riemannian Block Coordinate
Descent Method for Computing the Projection Robust Wasserstein
Distance</a>,
Proceedings of the 38th International Conference on Machine Learning
(ICML).</p>
<p>[33] Kerdoncuff T., Emonet R., Marc S. <aclass="reference external" href="https://hal.archives-ouvertes.fr/hal-03232509/document">Sampled Gromov
Wasserstein</a>,
Machine Learning Journal (MJL), 2021</p>
<p>[34] Feydy, J., Séjourné, T., Vialard, F. X., Amari, S. I., Trouvé, A.,
& Peyré, G. (2019, April). <aclass="reference external" href="http://proceedings.mlr.press/v89/feydy19a/feydy19a.pdf">Interpolating between optimal transport and
MMD using Sinkhorn
divergences</a>.
In The 22nd International Conference on Artificial Intelligence and
Statistics (pp. 2681-2690). PMLR.</p>
<p>[35] Deshpande, I., Hu, Y. T., Sun, R., Pyrros, A., Siddiqui, N.,
Koyejo, S., … & Schwing, A. G. (2019). <aclass="reference external" href="https://openaccess.thecvf.com/content_CVPR_2019/papers/Deshpande_Max-Sliced_Wasserstein_Distance_and_Its_Use_for_GANs_CVPR_2019_paper.pdf">Max-sliced wasserstein
distance and its use for
gans</a>.
In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern
Recognition (pp. 10648-10656).</p>
<p>[36] Liutkus, A., Simsekli, U., Majewski, S., Durmus, A., & Stöter, F.
R. (2019, May). <aclass="reference external" href="http://proceedings.mlr.press/v97/liutkus19a/liutkus19a.pdf">Sliced-Wasserstein flows: Nonparametric generative
modeling via optimal transport and
diffusions</a>.
In International Conference on Machine Learning (pp. 4104-4113). PMLR.</p>
<p>[37] Janati, H., Cuturi, M., Gramfort, A. <aclass="reference external" href="http://proceedings.mlr.press/v119/janati20a/janati20a.pdf">Debiased sinkhorn
barycenters</a>
Proceedings of the 37th International Conference on Machine Learning,
PMLR 119:4692-4701, 2020</p>
<p>[38] C. Vincent-Cuaz, T. Vayer, R. Flamary, M. Corneli, N. Courty,
<aclass="reference external" href="https://arxiv.org/pdf/2102.06555.pdf">Online Graph Dictionary
Learning</a>, International
Conference on Machine Learning (ICML), 2021.</p>
</section>
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