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<!DOCTYPE html>
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POT Python Optimal Transport
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<li class="toctree-l1 current"><a class="current reference internal" href="#">API and modules</a><ul>
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<li class="toctree-l2"><a class="reference internal" href="#module-ot">Main <code class="xref py py-mod docutils literal notranslate"><span class="pre">ot</span></code> functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="#ot.barycenter"><code class="docutils literal notranslate"><span class="pre">barycenter()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.barycenter_unbalanced"><code class="docutils literal notranslate"><span class="pre">barycenter_unbalanced()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.binary_search_circle"><code class="docutils literal notranslate"><span class="pre">binary_search_circle()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.dist"><code class="docutils literal notranslate"><span class="pre">dist()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.dist_batch"><code class="docutils literal notranslate"><span class="pre">dist_batch()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.emd"><code class="docutils literal notranslate"><span class="pre">emd()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.emd2"><code class="docutils literal notranslate"><span class="pre">emd2()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.emd2_1d"><code class="docutils literal notranslate"><span class="pre">emd2_1d()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.emd_1d"><code class="docutils literal notranslate"><span class="pre">emd_1d()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.factored_optimal_transport"><code class="docutils literal notranslate"><span class="pre">factored_optimal_transport()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.fused_gromov_wasserstein"><code class="docutils literal notranslate"><span class="pre">fused_gromov_wasserstein()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.fused_gromov_wasserstein2"><code class="docutils literal notranslate"><span class="pre">fused_gromov_wasserstein2()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.gromov_barycenters"><code class="docutils literal notranslate"><span class="pre">gromov_barycenters()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.gromov_wasserstein"><code class="docutils literal notranslate"><span class="pre">gromov_wasserstein()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.gromov_wasserstein2"><code class="docutils literal notranslate"><span class="pre">gromov_wasserstein2()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.linear_circular_ot"><code class="docutils literal notranslate"><span class="pre">linear_circular_ot()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.linear_sliced_wasserstein_sphere"><code class="docutils literal notranslate"><span class="pre">linear_sliced_wasserstein_sphere()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.lowrank_gromov_wasserstein_samples"><code class="docutils literal notranslate"><span class="pre">lowrank_gromov_wasserstein_samples()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.lowrank_sinkhorn"><code class="docutils literal notranslate"><span class="pre">lowrank_sinkhorn()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.max_sliced_wasserstein_distance"><code class="docutils literal notranslate"><span class="pre">max_sliced_wasserstein_distance()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.semidiscrete_wasserstein2_unif_circle"><code class="docutils literal notranslate"><span class="pre">semidiscrete_wasserstein2_unif_circle()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.sinkhorn"><code class="docutils literal notranslate"><span class="pre">sinkhorn()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.sinkhorn2"><code class="docutils literal notranslate"><span class="pre">sinkhorn2()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.sinkhorn_lpl1_mm"><code class="docutils literal notranslate"><span class="pre">sinkhorn_lpl1_mm()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.sinkhorn_unbalanced"><code class="docutils literal notranslate"><span class="pre">sinkhorn_unbalanced()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.sinkhorn_unbalanced2"><code class="docutils literal notranslate"><span class="pre">sinkhorn_unbalanced2()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.sliced_wasserstein_distance"><code class="docutils literal notranslate"><span class="pre">sliced_wasserstein_distance()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.sliced_wasserstein_sphere"><code class="docutils literal notranslate"><span class="pre">sliced_wasserstein_sphere()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.sliced_wasserstein_sphere_unif"><code class="docutils literal notranslate"><span class="pre">sliced_wasserstein_sphere_unif()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.solve"><code class="docutils literal notranslate"><span class="pre">solve()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.solve_batch"><code class="docutils literal notranslate"><span class="pre">solve_batch()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.solve_gromov"><code class="docutils literal notranslate"><span class="pre">solve_gromov()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.solve_gromov_batch"><code class="docutils literal notranslate"><span class="pre">solve_gromov_batch()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.solve_sample"><code class="docutils literal notranslate"><span class="pre">solve_sample()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.solve_sample_batch"><code class="docutils literal notranslate"><span class="pre">solve_sample_batch()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.tic"><code class="docutils literal notranslate"><span class="pre">tic()</span></code></a></li>
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<li class="toctree-l3"><a class="reference internal" href="#ot.wasserstein_1d"><code class="docutils literal notranslate"><span class="pre">wasserstein_1d()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.wasserstein_circle"><code class="docutils literal notranslate"><span class="pre">wasserstein_circle()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#ot.weak_optimal_transport"><code class="docutils literal notranslate"><span class="pre">weak_optimal_transport()</span></code></a></li>
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<section id="api-and-modules">
<span id="sphx-glr-api-reference"></span><h1>API and modules<a class="headerlink" href="#api-and-modules" title="Link to this heading"></a></h1>
<p><a class="reference internal" href="#module-ot" title="ot"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot</span></code></a>:</p>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="gen_modules/ot.backend.html#module-ot.backend" title="ot.backend"><code class="xref py py-obj docutils literal notranslate"><span class="pre">backend</span></code></a></p></td>
<td><p>Multi-lib backend for POT</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="gen_modules/ot.batch.html#module-ot.batch" title="ot.batch"><code class="xref py py-obj docutils literal notranslate"><span class="pre">batch</span></code></a></p></td>
<td><p>Batch operations for optimal transport.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="gen_modules/ot.bregman.html#module-ot.bregman" title="ot.bregman"><code class="xref py py-obj docutils literal notranslate"><span class="pre">bregman</span></code></a></p></td>
<td><p>Solvers related to Bregman projections for entropic regularized OT</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="gen_modules/ot.coot.html#module-ot.coot" title="ot.coot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">coot</span></code></a></p></td>
<td><p>CO-Optimal Transport solver</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="gen_modules/ot.da.html#module-ot.da" title="ot.da"><code class="xref py py-obj docutils literal notranslate"><span class="pre">da</span></code></a></p></td>
<td><p>Domain adaptation with optimal transport</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="gen_modules/ot.datasets.html#module-ot.datasets" title="ot.datasets"><code class="xref py py-obj docutils literal notranslate"><span class="pre">datasets</span></code></a></p></td>
<td><p>Simple example datasets</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="gen_modules/ot.dr.html#module-ot.dr" title="ot.dr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">dr</span></code></a></p></td>
<td><p>Dimension reduction with OT</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="gen_modules/ot.factored.html#module-ot.factored" title="ot.factored"><code class="xref py py-obj docutils literal notranslate"><span class="pre">factored</span></code></a></p></td>
<td><p>Factored OT solvers (low rank, cost or OT plan)</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="gen_modules/ot.gaussian.html#module-ot.gaussian" title="ot.gaussian"><code class="xref py py-obj docutils literal notranslate"><span class="pre">gaussian</span></code></a></p></td>
<td><p>Optimal transport for Gaussian distributions</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="gen_modules/ot.gmm.html#module-ot.gmm" title="ot.gmm"><code class="xref py py-obj docutils literal notranslate"><span class="pre">gmm</span></code></a></p></td>
<td><p>Optimal transport for Gaussian Mixtures</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="gen_modules/ot.gnn.html#module-ot.gnn" title="ot.gnn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">gnn</span></code></a></p></td>
<td><p>Layers and functions for optimal transport in Graph Neural Networks.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="gen_modules/ot.gromov.html#module-ot.gromov" title="ot.gromov"><code class="xref py py-obj docutils literal notranslate"><span class="pre">gromov</span></code></a></p></td>
<td><p>Solvers related to Gromov-Wasserstein problems.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="gen_modules/ot.lowrank.html#module-ot.lowrank" title="ot.lowrank"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lowrank</span></code></a></p></td>
<td><p>Low rank OT solvers</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="gen_modules/ot.lp.html#module-ot.lp" title="ot.lp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lp</span></code></a></p></td>
<td><p>Solvers for the original linear program OT problem.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="gen_modules/ot.mapping.html#module-ot.mapping" title="ot.mapping"><code class="xref py py-obj docutils literal notranslate"><span class="pre">mapping</span></code></a></p></td>
<td><p>Optimal Transport maps and variants</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="gen_modules/ot.optim.html#module-ot.optim" title="ot.optim"><code class="xref py py-obj docutils literal notranslate"><span class="pre">optim</span></code></a></p></td>
<td><p>Generic solvers for regularized OT or its semi-relaxed version.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="gen_modules/ot.partial.html#module-ot.partial" title="ot.partial"><code class="xref py py-obj docutils literal notranslate"><span class="pre">partial</span></code></a></p></td>
<td><p>Efficient 1D solver for the partial optimal transport problem.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="gen_modules/ot.plot.html#module-ot.plot" title="ot.plot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">plot</span></code></a></p></td>
<td><p>Functions for plotting OT matrices</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="gen_modules/ot.regpath.html#module-ot.regpath" title="ot.regpath"><code class="xref py py-obj docutils literal notranslate"><span class="pre">regpath</span></code></a></p></td>
<td><p>Regularization path OT solvers</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="gen_modules/ot.sliced.html#module-ot.sliced" title="ot.sliced"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sliced</span></code></a></p></td>
<td><p>Sliced OT Distances</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="gen_modules/ot.smooth.html#module-ot.smooth" title="ot.smooth"><code class="xref py py-obj docutils literal notranslate"><span class="pre">smooth</span></code></a></p></td>
<td><p>Smooth and Sparse (KL an L2 reg.) and sparsity-constrained OT solvers.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="gen_modules/ot.stochastic.html#module-ot.stochastic" title="ot.stochastic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">stochastic</span></code></a></p></td>
<td><p>Stochastic solvers for regularized OT.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="gen_modules/ot.unbalanced.html#module-ot.unbalanced" title="ot.unbalanced"><code class="xref py py-obj docutils literal notranslate"><span class="pre">unbalanced</span></code></a></p></td>
<td><p>Solvers related to Unbalanced Optimal Transport problems.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="gen_modules/ot.utils.html#module-ot.utils" title="ot.utils"><code class="xref py py-obj docutils literal notranslate"><span class="pre">utils</span></code></a></p></td>
<td><p>Various useful functions</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="gen_modules/ot.weak.html#module-ot.weak" title="ot.weak"><code class="xref py py-obj docutils literal notranslate"><span class="pre">weak</span></code></a></p></td>
<td><p>Weak optimal ransport solvers</p></td>
</tr>
</tbody>
</table>
<section id="module-ot">
<span id="main-ot-functions"></span><h2>Main <a class="reference internal" href="#module-ot" title="ot"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot</span></code></a> functions<a class="headerlink" href="#module-ot" title="Link to this heading"></a></h2>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>The list of automatically imported sub-modules is as follows:
<a class="reference internal" href="gen_modules/ot.lp.html#module-ot.lp" title="ot.lp"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot.lp</span></code></a>, <a class="reference internal" href="gen_modules/ot.bregman.html#module-ot.bregman" title="ot.bregman"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot.bregman</span></code></a>, <a class="reference internal" href="gen_modules/ot.optim.html#module-ot.optim" title="ot.optim"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot.optim</span></code></a>
<a class="reference internal" href="gen_modules/ot.utils.html#module-ot.utils" title="ot.utils"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot.utils</span></code></a>, <a class="reference internal" href="gen_modules/ot.datasets.html#module-ot.datasets" title="ot.datasets"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot.datasets</span></code></a>,
<a class="reference internal" href="gen_modules/ot.gromov.html#module-ot.gromov" title="ot.gromov"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot.gromov</span></code></a>, <a class="reference internal" href="gen_modules/ot.smooth.html#module-ot.smooth" title="ot.smooth"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot.smooth</span></code></a>
<a class="reference internal" href="gen_modules/ot.stochastic.html#module-ot.stochastic" title="ot.stochastic"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot.stochastic</span></code></a>, <a class="reference internal" href="gen_modules/ot.partial.html#module-ot.partial" title="ot.partial"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot.partial</span></code></a>, <a class="reference internal" href="gen_modules/ot.regpath.html#module-ot.regpath" title="ot.regpath"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot.regpath</span></code></a>
, <a class="reference internal" href="gen_modules/ot.unbalanced.html#module-ot.unbalanced" title="ot.unbalanced"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot.unbalanced</span></code></a>, <a class="reference internal" href="gen_modules/ot.mapping.html#module-ot.mapping" title="ot.mapping"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot.mapping</span></code></a> .
The following sub-modules are not imported due to additional dependencies:
- <a class="reference internal" href="gen_modules/ot.dr.html#module-ot.dr" title="ot.dr"><code class="xref any py py-mod docutils literal notranslate"><span class="pre">ot.dr</span></code></a> : depends on <code class="code docutils literal notranslate"><span class="pre">pymanopt</span></code> and <code class="code docutils literal notranslate"><span class="pre">autograd</span></code>.
- <a class="reference internal" href="gen_modules/ot.plot.html#module-ot.plot" title="ot.plot"><code class="xref any py py-mod docutils literal notranslate"><span class="pre">ot.plot</span></code></a> : depends on <code class="code docutils literal notranslate"><span class="pre">matplotlib</span></code></p>
</div>
<dl class="py function">
<dt class="sig sig-object py" id="ot.barycenter">
<span class="sig-prename descclassname"><span class="pre">ot.</span></span><span class="sig-name descname"><span class="pre">barycenter</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">M</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">reg</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">weights</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">method</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'sinkhorn'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">numItermax</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">10000</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">stopThr</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0001</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">verbose</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">warn</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/ot/bregman/_barycenter.html#barycenter"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.barycenter" title="Link to this definition"></a></dt>
<dd><p>Compute the entropic regularized wasserstein barycenter of distributions <span class="math notranslate nohighlight">\(\mathbf{A}\)</span></p>
<blockquote>
<div><p>The function solves the following optimization problem:</p>
</div></blockquote>
<div class="math notranslate nohighlight">
\[\mathbf{a} = \mathop{\arg \min}_\mathbf{a} \quad \sum_i W_{reg}(\mathbf{a},\mathbf{a}_i)\]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(W_{reg}(\cdot,\cdot)\)</span> is the entropic regularized Wasserstein
distance (see <a class="reference internal" href="gen_modules/ot.bregman.html#ot.bregman.sinkhorn" title="ot.bregman.sinkhorn"><code class="xref py py-func docutils literal notranslate"><span class="pre">ot.bregman.sinkhorn()</span></code></a>)
if <cite>method</cite> is <cite>sinkhorn</cite> or <cite>sinkhorn_stabilized</cite> or <cite>sinkhorn_log</cite>.</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{a}_i\)</span> are training distributions in the columns of matrix
<span class="math notranslate nohighlight">\(\mathbf{A}\)</span></p></li>
<li><p><cite>reg</cite> and <span class="math notranslate nohighlight">\(\mathbf{M}\)</span> are respectively the regularization term and
the cost matrix for OT</p></li>
</ul>
<p>The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling
algorithm as proposed in <a class="reference internal" href="gen_modules/ot.bregman.html#references-barycenter"><span class="std std-ref">[3]</span></a></p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>A</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>dim</em><em>, </em><em>n_hists</em><em>)</em>) – <cite>n_hists</cite> training distributions <span class="math notranslate nohighlight">\(\mathbf{a}_i\)</span> of size <cite>dim</cite></p></li>
<li><p><strong>M</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>dim</em><em>, </em><em>dim</em><em>)</em>) – loss matrix for OT</p></li>
<li><p><strong>reg</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a>) – Regularization term > 0</p></li>
<li><p><strong>method</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.13)"><em>str</em></a><em> (</em><em>optional</em><em>)</em>) – method used for the solver either ‘sinkhorn’ or ‘sinkhorn_stabilized’ or ‘sinkhorn_log’</p></li>
<li><p><strong>weights</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_hists</em><em>,</em><em>)</em>) – Weights of each histogram <span class="math notranslate nohighlight">\(\mathbf{a}_i\)</span> on the simplex (barycentric coordinates)</p></li>
<li><p><strong>numItermax</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.13)"><em>int</em></a><em>, </em><em>optional</em>) – Max number of iterations</p></li>
<li><p><strong>stopThr</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em>) – Stop threshold on error (>0)</p></li>
<li><p><strong>verbose</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em>) – Print information along iterations</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em>) – record log if True</p></li>
<li><p><strong>warn</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em>) – if True, raises a warning if the algorithm doesn’t convergence.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>a</strong> (<em>(dim,) array-like</em>) – Wasserstein barycenter</p></li>
<li><p><strong>log</strong> (<em>dict</em>) – log dictionary return only if log==True in parameters</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric" id="references-barycenter">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id1" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></span>
<p>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.</p>
</aside>
</aside>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.barycenter_unbalanced">
<span class="sig-prename descclassname"><span class="pre">ot.</span></span><span class="sig-name descname"><span class="pre">barycenter_unbalanced</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">M</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">reg</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">reg_m</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">method</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'sinkhorn'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">weights</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">numItermax</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1000</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">stopThr</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-06</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">verbose</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/ot/unbalanced/_sinkhorn.html#barycenter_unbalanced"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.barycenter_unbalanced" title="Link to this definition"></a></dt>
<dd><p>Compute the entropic unbalanced wasserstein barycenter of <span class="math notranslate nohighlight">\(\mathbf{A}\)</span>.</p>
<blockquote>
<div><p>The function solves the following optimization problem with <span class="math notranslate nohighlight">\(\mathbf{a}\)</span></p>
</div></blockquote>
<div class="math notranslate nohighlight">
\[\mathbf{a} = \mathop{\arg \min}_\mathbf{a} \quad \sum_i W_{u_{reg}}(\mathbf{a},\mathbf{a}_i)\]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(W_{u_{reg}}(\cdot,\cdot)\)</span> is the unbalanced entropic regularized Wasserstein distance (see <a class="reference internal" href="gen_modules/ot.unbalanced.html#ot.unbalanced.sinkhorn_unbalanced" title="ot.unbalanced.sinkhorn_unbalanced"><code class="xref py py-func docutils literal notranslate"><span class="pre">ot.unbalanced.sinkhorn_unbalanced()</span></code></a>)</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{a}_i\)</span> are training distributions in the columns of matrix <span class="math notranslate nohighlight">\(\mathbf{A}\)</span></p></li>
<li><p>reg and <span class="math notranslate nohighlight">\(\mathbf{M}\)</span> are respectively the regularization term and the cost matrix for OT</p></li>
<li><p>reg_mis the marginal relaxation hyperparameter</p></li>
</ul>
<p>The algorithm used for solving the problem is the generalized
Sinkhorn-Knopp matrix scaling algorithm as proposed in <a class="reference internal" href="gen_modules/ot.unbalanced.html#references-barycenter-unbalanced"><span class="std std-ref">[10]</span></a></p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>A</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>dim</em><em>, </em><em>n_hists</em><em>)</em>) – <cite>n_hists</cite> training distributions <span class="math notranslate nohighlight">\(\mathbf{a}_i\)</span> of dimension <cite>dim</cite></p></li>
<li><p><strong>M</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>dim</em><em>, </em><em>dim</em><em>)</em>) – ground metric matrix for OT.</p></li>
<li><p><strong>reg</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a>) – Entropy regularization term > 0</p></li>
<li><p><strong>reg_m</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a>) – Marginal relaxation term > 0</p></li>
<li><p><strong>weights</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_hists</em><em>,</em><em>) </em><em>optional</em>) – Weight of each distribution (barycentric coordinates)
If None, uniform weights are used.</p></li>
<li><p><strong>numItermax</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.13)"><em>int</em></a><em>, </em><em>optional</em>) – Max number of iterations</p></li>
<li><p><strong>stopThr</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em>) – Stop threshold on error (> 0)</p></li>
<li><p><strong>verbose</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em>) – Print information along iterations</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em>) – record log if True</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>a</strong> (<em>array-like, shape (dim,)</em>) – Unbalanced Wasserstein barycenter</p></li>
<li><p><strong>log</strong> (<em>dict</em>) – log dictionary return only if log==True in parameters</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric" id="references-barycenter-unbalanced">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id2" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></span>
<p>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.</p>
</aside>
<aside class="footnote brackets" id="id3" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>10<span class="fn-bracket">]</span></span>
<p>Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016).
Scaling algorithms for unbalanced transport problems. arXiv preprin
arXiv:1607.05816.</p>
</aside>
</aside>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.binary_search_circle">
<span class="sig-prename descclassname"><span class="pre">ot.</span></span><span class="sig-name descname"><span class="pre">binary_search_circle</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u_values</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">v_values</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">u_weights</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">v_weights</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Lm</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">10</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Lp</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">10</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">tm</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">-1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">tp</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eps</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-06</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">require_sort</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/ot/lp/solver_1d.html#binary_search_circle"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.binary_search_circle" title="Link to this definition"></a></dt>
<dd><p>Computes the Wasserstein distance on the circle using the Binary search algorithm proposed in [44].
Samples need to be in <span class="math notranslate nohighlight">\(S^1\cong [0,1[\)</span>. If they are on <span class="math notranslate nohighlight">\(\mathbb{R}\)</span>,
takes the value modulo 1.
If the values are on <span class="math notranslate nohighlight">\(S^1\subset\mathbb{R}^2\)</span>, it is required to first find the coordinates
using e.g. the atan2 function.</p>
<div class="math notranslate nohighlight">
\[W_p^p(u,v) = \inf_{\theta\in\mathbb{R}}\int_0^1 |F_u^{-1}(q) - (F_v-\theta)^{-1}(q)|^p\ \mathrm{d}q\]</div>
<p>where:</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(F_u\)</span> and <span class="math notranslate nohighlight">\(F_v\)</span> are respectively the cdfs of <span class="math notranslate nohighlight">\(u\)</span> and <span class="math notranslate nohighlight">\(v\)</span></p></li>
</ul>
<p>For values <span class="math notranslate nohighlight">\(x=(x_1,x_2)\in S^1\)</span>, it is required to first get their coordinates with</p>
<div class="math notranslate nohighlight">
\[u = \frac{\pi + \mathrm{atan2}(-x_2,-x_1)}{2\pi}\]</div>
<p>using e.g. ot.utils.get_coordinate_circle(x)</p>
<p>The function runs on backend but tensorflow and jax are not supported.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>u_values</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>...</em><em>)</em>) – samples in the source domain (coordinates on [0,1[)</p></li>
<li><p><strong>v_values</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>...</em><em>)</em>) – samples in the target domain (coordinates on [0,1[)</p></li>
<li><p><strong>u_weights</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>...</em><em>)</em><em>, </em><em>optional</em>) – samples weights in the source domain</p></li>
<li><p><strong>v_weights</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>...</em><em>)</em><em>, </em><em>optional</em>) – samples weights in the target domain</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em><em> (</em><em>default=1</em><em>)</em>) – Power p used for computing the Wasserstein distance</p></li>
<li><p><strong>Lm</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.13)"><em>int</em></a><em>, </em><em>optional</em>) – Lower bound dC</p></li>
<li><p><strong>Lp</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.13)"><em>int</em></a><em>, </em><em>optional</em>) – Upper bound dC</p></li>
<li><p><strong>tm</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em>) – Lower bound theta</p></li>
<li><p><strong>tp</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em>) – Upper bound theta</p></li>
<li><p><strong>eps</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em>) – Stopping condition</p></li>
<li><p><strong>require_sort</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em>) – If True, sort the values.</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em>) – If True, returns also the optimal theta</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>loss</strong> (<em>float/array-like, shape (…)</em>) – Batched cost associated to the optimal transportation</p></li>
<li><p><strong>log</strong> (<em>dict, optional</em>) – log dictionary returned only if log==True in parameters</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">u</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.8</span><span class="p">]])</span><span class="o">%</span><span class="mi">1</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.4</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.7</span><span class="p">]])</span><span class="o">%</span><span class="mi">1</span>
<span class="gp">>>> </span><span class="n">binary_search_circle</span><span class="p">(</span><span class="n">u</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">v</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="go">array([0.1])</span>
</pre></div>
</div>
<p class="rubric">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id4" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>44<span class="fn-bracket">]</span></span>
<p>Delon, Julie, Julien Salomon, and Andrei Sobolevski. “Fast transport optimization for Monge costs on the circle.” SIAM Journal on Applied Mathematics 70.7 (2010): 2239-2258.</p>
</aside>
</aside>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.dist">
<span class="sig-prename descclassname"><span class="pre">ot.</span></span><span class="sig-name descname"><span class="pre">dist</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x2</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">metric</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'sqeuclidean'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">w</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">backend</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'auto'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">use_tensor</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/ot/utils.html#dist"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.dist" title="Link to this definition"></a></dt>
<dd><p>Compute distance between samples in <span class="math notranslate nohighlight">\(\mathbf{x_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{x_2}\)</span></p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function is backend-compatible and will work on arrays
from all compatible backends for the following metrics:
‘sqeuclidean’, ‘euclidean’, ‘cityblock’, ‘minkowski’, ‘cosine’, ‘correlation’.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>x1</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n1</em><em>,</em><em>d</em><em>)</em>) – matrix with <cite>n1</cite> samples of size <cite>d</cite></p></li>
<li><p><strong>x2</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n2</em><em>,</em><em>d</em><em>)</em><em>, </em><em>optional</em>) – matrix with <cite>n2</cite> samples of size <cite>d</cite> (if None then <span class="math notranslate nohighlight">\(\mathbf{x_2} = \mathbf{x_1}\)</span>)</p></li>
<li><p><strong>metric</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.13)"><em>str</em></a><em> | </em><em>callable</em><em>, </em><em>optional</em>) – ‘sqeuclidean’ or ‘euclidean’ on all backends. On numpy the function also
accepts from the scipy.spatial.distance.cdist function : ‘braycurtis’,
‘canberra’, ‘chebyshev’, ‘cityblock’, ‘correlation’, ‘cosine’, ‘dice’,
‘euclidean’, ‘hamming’, ‘jaccard’, ‘kulczynski1’, ‘mahalanobis’,
‘matching’, ‘minkowski’, ‘rogerstanimoto’, ‘russellrao’, ‘seuclidean’,
‘sokalmichener’, ‘sokalsneath’, ‘sqeuclidean’, ‘wminkowski’, ‘yule’.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em>) – p-norm for the Minkowski and the Weighted Minkowski metrics. Default value is 2.</p></li>
<li><p><strong>w</strong> (<em>array-like</em><em>, </em><em>rank 1</em>) – Weights for the weighted metrics.</p></li>
<li><p><strong>backend</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.13)"><em>str</em></a><em>, </em><em>optional</em>) – Backend to use for the computation. If ‘auto’, the backend is
automatically selected based on the input data. if ‘scipy’,
the <code class="docutils literal notranslate"><span class="pre">scipy.spatial.distance.cdist</span></code> function is used (and gradients are
detached).</p></li>
<li><p><strong>use_tensor</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em>) – If true use tensorized computation for the distance matrix which can
cause memory issues for large datasets. Default is False and the
parameter is used only for the ‘cityblock’ and ‘minkowski’ metrics.</p></li>
<li><p><strong>nx</strong> (<a class="reference internal" href="gen_modules/ot.backend.html#id0" title="ot.backend.Backend"><em>Backend</em></a><em>, </em><em>optional</em>) – Backend to perform computations on. If omitted, the backend defaults to that of <cite>x1</cite>.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>M</strong> – distance matrix computed with given metric</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>array-like, shape (<cite>n1</cite>, <cite>n2</cite>)</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.dist_batch">
<span class="sig-prename descclassname"><span class="pre">ot.</span></span><span class="sig-name descname"><span class="pre">dist_batch</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">X2</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">metric</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'sqeuclidean'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/ot/batch/_linear.html#dist_batch"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.dist_batch" title="Link to this definition"></a></dt>
<dd><p>Batched version of ot.dist, use it to compute many distance matrices in parallel.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>X1</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>b</em><em>,</em><em>n1</em><em>,</em><em>d</em><em>)</em>) – <cite>b</cite> matrices with <cite>n1</cite> samples of size <cite>d</cite></p></li>
<li><p><strong>X2</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>b</em><em>,</em><em>n2</em><em>,</em><em>d</em><em>)</em><em>, </em><em>optional</em>) – <cite>b</cite> matrices with <cite>n2</cite> samples of size <cite>d</cite> (if None then <span class="math notranslate nohighlight">\(\mathbf{X_2} = \mathbf{X_1}\)</span>)</p></li>
<li><p><strong>metric</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.13)"><em>str</em></a><em>, </em><em>optional</em>) – ‘sqeuclidean’, ‘euclidean’, ‘minkowski’ or ‘kl’</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em>) – p-norm for the Minkowski metrics. Default value is 2.</p></li>
<li><p><strong>nx</strong> (<a class="reference internal" href="gen_modules/ot.backend.html#id0" title="ot.backend.Backend"><em>Backend</em></a><em>, </em><em>optional</em>) – Backend to perform computations on. If omitted, the backend defaults to that of <cite>x1</cite>.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>M</strong> – distance matrix computed with given metric</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>array-like, shape (<cite>b</cite>, <cite>n1</cite>, <cite>n2</cite>)</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">ot.batch</span><span class="w"> </span><span class="kn">import</span> <span class="n">dist_batch</span>
<span class="gp">>>> </span><span class="n">X1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">dist_batch</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">,</span> <span class="n">metric</span><span class="o">=</span><span class="s2">"euclidean"</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">M</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(5, 10, 15)</span>
</pre></div>
</div>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="#ot.dist" title="ot.dist"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ot.dist</span></code></a></dt><dd><p>equivalent non-batched function.</p>
</dd>
</dl>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.emd">
<span class="sig-prename descclassname"><span class="pre">ot.</span></span><span class="sig-name descname"><span class="pre">emd</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">M</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">numItermax</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">100000</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">center_dual</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">numThreads</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">check_marginals</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/ot/lp/_network_simplex.html#emd"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.emd" title="Link to this definition"></a></dt>
<dd><p>Solves the Earth Movers distance problem and returns the OT matrix</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F\\s.t. \ \gamma \mathbf{1} = \mathbf{a}\\ \gamma^T \mathbf{1} = \mathbf{b}\\ \gamma \geq 0\end{aligned}\end{align} \]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\mathbf{M}\)</span> is the metric cost matrix</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{a}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> are the sample weights</p></li>
</ul>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>Note that the <span class="math notranslate nohighlight">\(\mathbf{M}\)</span> matrix in numpy needs to be a C-order
numpy.array in float64 format. It will be converted if not in this
format</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function is backend-compatible and will work on arrays
from all compatible backends. But the algorithm uses the C++ CPU backend
which can lead to copy overhead on GPU arrays.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function will cast the computed transport plan to the data type
of the provided input with the following priority: <span class="math notranslate nohighlight">\(\mathbf{a}\)</span>,
then <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>, then <span class="math notranslate nohighlight">\(\mathbf{M}\)</span> if marginals are not provided.
Casting to an integer tensor might result in a loss of precision.
If this behaviour is unwanted, please make sure to provide a
floating point input.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>An error will be raised if the vectors <span class="math notranslate nohighlight">\(\mathbf{a}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> do not sum to the same value.</p>
</div>
<p>Uses the algorithm proposed in <a class="reference internal" href="gen_modules/ot.lp.html#references-emd"><span class="std std-ref">[1]</span></a>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>a</strong> (<em>(</em><em>ns</em><em>,</em><em>) </em><em>array-like</em><em>, </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a>) – Source histogram (uniform weight if empty list)</p></li>
<li><p><strong>b</strong> (<em>(</em><em>nt</em><em>,</em><em>) </em><em>array-like</em><em>, </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a>) – Target histogram (uniform weight if empty list)</p></li>
<li><p><strong>M</strong> (<em>(</em><em>ns</em><em>,</em><em>nt</em><em>) </em><em>array-like</em><em>, </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a>) – Loss matrix (c-order array in numpy with type float64)</p></li>
<li><p><strong>numItermax</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.13)"><em>int</em></a><em>, </em><em>optional</em><em> (</em><em>default=100000</em><em>)</em>) – The maximum number of iterations before stopping the optimization
algorithm if it has not converged.</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em><em> (</em><em>default=False</em><em>)</em>) – If True, returns a dictionary containing the cost and dual variables.
Otherwise returns only the optimal transportation matrix.</p></li>
<li><p><strong>center_dual</strong> (<em>boolean</em><em>, </em><em>optional</em><em> (</em><em>default=True</em><em>)</em>) – If True, centers the dual potential using function
<code class="xref py py-func docutils literal notranslate"><span class="pre">ot.lp.center_ot_dual()</span></code>.</p></li>
<li><p><strong>numThreads</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.13)"><em>int</em></a><em> or </em><em>"max"</em><em>, </em><em>optional</em><em> (</em><em>default=1</em><em>, </em><em>i.e. OpenMP is not used</em><em>)</em>) – If compiled with OpenMP, chooses the number of threads to parallelize.
“max” selects the highest number possible.</p></li>
<li><p><strong>check_marginals</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em><em> (</em><em>default=True</em><em>)</em>) – If True, checks that the marginals mass are equal. If False, skips the
check.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>gamma</strong> (<em>array-like, shape (ns, nt)</em>) – Optimal transportation matrix for the given
parameters</p></li>
<li><p><strong>log</strong> (<em>dict, optional</em>) – If input log is true, a dictionary containing the
cost and dual variables and exit status</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<p>Simple example with obvious solution. The function emd accepts lists and
perform automatic conversion to numpy arrays</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">ot</span>
<span class="gp">>>> </span><span class="n">a</span><span class="o">=</span><span class="p">[</span><span class="mf">.5</span><span class="p">,</span><span class="mf">.5</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">b</span><span class="o">=</span><span class="p">[</span><span class="mf">.5</span><span class="p">,</span><span class="mf">.5</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">M</span><span class="o">=</span><span class="p">[[</span><span class="mf">0.</span><span class="p">,</span><span class="mf">1.</span><span class="p">],[</span><span class="mf">1.</span><span class="p">,</span><span class="mf">0.</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">ot</span><span class="o">.</span><span class="n">emd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">M</span><span class="p">)</span>
<span class="go">array([[0.5, 0. ],</span>
<span class="go"> [0. , 0.5]])</span>
</pre></div>
</div>
<p class="rubric" id="references-emd">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id5" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></span>
<p>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.</p>
</aside>
</aside>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="gen_modules/ot.bregman.html#ot.bregman.sinkhorn" title="ot.bregman.sinkhorn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ot.bregman.sinkhorn</span></code></a></dt><dd><p>Entropic regularized OT</p>
</dd>
<dt><a class="reference internal" href="gen_modules/ot.optim.html#id0" title="ot.optim.cg"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ot.optim.cg</span></code></a></dt><dd><p>General regularized OT</p>
</dd>
</dl>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.emd2">
<span class="sig-prename descclassname"><span class="pre">ot.</span></span><span class="sig-name descname"><span class="pre">emd2</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">M</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">processes</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">numItermax</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">100000</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">return_matrix</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">center_dual</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">numThreads</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">check_marginals</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/ot/lp/_network_simplex.html#emd2"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.emd2" title="Link to this definition"></a></dt>
<dd><p>Solves the Earth Movers distance problem and returns the loss</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\min_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F\\s.t. \ \gamma \mathbf{1} = \mathbf{a}\\ \gamma^T \mathbf{1} = \mathbf{b}\\ \gamma \geq 0\end{aligned}\end{align} \]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\mathbf{M}\)</span> is the metric cost matrix</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{a}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> are the sample weights</p></li>
</ul>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function is backend-compatible and will work on arrays
from all compatible backends. But the algorithm uses the C++ CPU backend
which can lead to copy overhead on GPU arrays.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function will cast the computed transport plan and
transportation loss to the data type of the provided input with the
following priority: <span class="math notranslate nohighlight">\(\mathbf{a}\)</span>, then <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>,
then <span class="math notranslate nohighlight">\(\mathbf{M}\)</span> if marginals are not provided.
Casting to an integer tensor might result in a loss of precision.
If this behaviour is unwanted, please make sure to provide a
floating point input.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>An error will be raised if the vectors <span class="math notranslate nohighlight">\(\mathbf{a}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> do not sum to the same value.</p>
</div>
<p>Uses the algorithm proposed in <a class="reference internal" href="gen_modules/ot.lp.html#references-emd2"><span class="std std-ref">[1]</span></a>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>a</strong> (<em>(</em><em>ns</em><em>,</em><em>) </em><em>array-like</em><em>, </em><em>float64</em>) – Source histogram (uniform weight if empty list)</p></li>
<li><p><strong>b</strong> (<em>(</em><em>nt</em><em>,</em><em>) </em><em>array-like</em><em>, </em><em>float64</em>) – Target histogram (uniform weight if empty list)</p></li>
<li><p><strong>M</strong> (<em>(</em><em>ns</em><em>,</em><em>nt</em><em>) </em><em>array-like</em><em>, </em><em>float64</em>) – Loss matrix (for numpy c-order array with type float64)</p></li>
<li><p><strong>processes</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.13)"><em>int</em></a><em>, </em><em>optional</em><em> (</em><em>default=1</em><em>)</em>) – Nb of processes used for multiple emd computation (deprecated)</p></li>
<li><p><strong>numItermax</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.13)"><em>int</em></a><em>, </em><em>optional</em><em> (</em><em>default=100000</em><em>)</em>) – The maximum number of iterations before stopping the optimization
algorithm if it has not converged.</p></li>
<li><p><strong>log</strong> (<em>boolean</em><em>, </em><em>optional</em><em> (</em><em>default=False</em><em>)</em>) – If True, returns a dictionary containing dual
variables. Otherwise returns only the optimal transportation cost.</p></li>
<li><p><strong>return_matrix</strong> (<em>boolean</em><em>, </em><em>optional</em><em> (</em><em>default=False</em><em>)</em>) – If True, returns the optimal transportation matrix in the log.</p></li>
<li><p><strong>center_dual</strong> (<em>boolean</em><em>, </em><em>optional</em><em> (</em><em>default=True</em><em>)</em>) – If True, centers the dual potential using function
<code class="xref py py-func docutils literal notranslate"><span class="pre">ot.lp.center_ot_dual()</span></code>.</p></li>
<li><p><strong>numThreads</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.13)"><em>int</em></a><em> or </em><em>"max"</em><em>, </em><em>optional</em><em> (</em><em>default=1</em><em>, </em><em>i.e. OpenMP is not used</em><em>)</em>) – If compiled with OpenMP, chooses the number of threads to parallelize.
“max” selects the highest number possible.</p></li>
<li><p><strong>check_marginals</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em><em> (</em><em>default=True</em><em>)</em>) – If True, checks that the marginals mass are equal. If False, skips the
check.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>W</strong> (<em>float, array-like</em>) – Optimal transportation loss for the given parameters</p></li>
<li><p><strong>log</strong> (<em>dict</em>) – If input log is true, a dictionary containing dual
variables and exit status</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<p>Simple example with obvious solution. The function emd accepts lists and
perform automatic conversion to numpy arrays</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">ot</span>
<span class="gp">>>> </span><span class="n">a</span><span class="o">=</span><span class="p">[</span><span class="mf">.5</span><span class="p">,</span><span class="mf">.5</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">b</span><span class="o">=</span><span class="p">[</span><span class="mf">.5</span><span class="p">,</span><span class="mf">.5</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">M</span><span class="o">=</span><span class="p">[[</span><span class="mf">0.</span><span class="p">,</span><span class="mf">1.</span><span class="p">],[</span><span class="mf">1.</span><span class="p">,</span><span class="mf">0.</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">ot</span><span class="o">.</span><span class="n">emd2</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">M</span><span class="p">)</span>
<span class="go">0.0</span>
</pre></div>
</div>
<p class="rubric" id="references-emd2">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id6" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></span>
<p>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.</p>
</aside>
</aside>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="gen_modules/ot.bregman.html#ot.bregman.sinkhorn" title="ot.bregman.sinkhorn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ot.bregman.sinkhorn</span></code></a></dt><dd><p>Entropic regularized OT</p>
</dd>
<dt><a class="reference internal" href="gen_modules/ot.optim.html#id0" title="ot.optim.cg"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ot.optim.cg</span></code></a></dt><dd><p>General regularized OT</p>
</dd>
</dl>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.emd2_1d">
<span class="sig-prename descclassname"><span class="pre">ot.</span></span><span class="sig-name descname"><span class="pre">emd2_1d</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x_a</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x_b</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">metric</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'sqeuclidean'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dense</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/ot/lp/solver_1d.html#emd2_1d"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.emd2_1d" title="Link to this definition"></a></dt>
<dd><p>Solves the Earth Movers distance problem between 1d measures and returns
the loss</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\gamma = arg\min_\gamma \sum_i \sum_j \gamma_{ij} d(x_a[i], x_b[j])\\s.t. \gamma 1 = a,
\gamma^T 1= b,
\gamma\geq 0\end{aligned}\end{align} \]</div>
<p>where :</p>
<ul class="simple">
<li><p>d is the metric</p></li>
<li><p><span class="math notranslate nohighlight">\(x_a\)</span> and <span class="math notranslate nohighlight">\(x_b\)</span> are the samples</p></li>
<li><p>a and b are the sample weights</p></li>
</ul>
<p>This implementation only supports metrics
of the form <span class="math notranslate nohighlight">\(d(x, y) = |x - y|^p\)</span>.</p>
<p>Uses the algorithm detailed in <a href="#id85"><span class="problematic" id="id7">[1]_</span></a></p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>x_a</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>,</em><em>) or </em><em>(</em><em>ns</em><em>, </em><em>1</em><em>)</em>) – Source dirac locations (on the real line)</p></li>
<li><p><strong>x_b</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>nt</em><em>,</em><em>) or </em><em>(</em><em>ns</em><em>, </em><em>1</em><em>)</em>) – Target dirac locations (on the real line)</p></li>
<li><p><strong>a</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Source histogram (default is uniform weight)</p></li>
<li><p><strong>b</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>nt</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Target histogram (default is uniform weight)</p></li>
<li><p><strong>metric</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.13)"><em>str</em></a><em>, </em><em>optional</em><em> (</em><em>default='sqeuclidean'</em><em>)</em>) – Metric to be used. Only works with either of the strings
<cite>‘sqeuclidean’</cite>, <cite>‘minkowski’</cite>, <cite>‘cityblock’</cite>, or <cite>‘euclidean’</cite>.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em><em> (</em><em>default=1.0</em><em>)</em>) – The p-norm to apply for if metric=’minkowski’</p></li>
<li><p><strong>dense</strong> (<em>boolean</em><em>, </em><em>optional</em><em> (</em><em>default=True</em><em>)</em>) – If True, returns <span class="math notranslate nohighlight">\(\gamma\)</span> as a dense ndarray of shape (ns, nt).
Otherwise returns a sparse representation using scipy’s <cite>coo_matrix</cite>
format. Only used if log is set to True. Due to implementation details,
this function runs faster when dense is set to False.</p></li>
<li><p><strong>log</strong> (<em>boolean</em><em>, </em><em>optional</em><em> (</em><em>default=False</em><em>)</em>) – If True, returns a dictionary containing the transportation matrix.
Otherwise returns only the loss.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>loss</strong> (<em>float</em>) – Cost associated to the optimal transportation</p></li>
<li><p><strong>log</strong> (<em>dict</em>) – If input log is True, a dictionary containing the Optimal transportation
matrix for the given parameters</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<p>Simple example with obvious solution. The function emd2_1d accepts lists and
performs automatic conversion to numpy arrays</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">ot</span>
<span class="gp">>>> </span><span class="n">a</span><span class="o">=</span><span class="p">[</span><span class="mf">.5</span><span class="p">,</span> <span class="mf">.5</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">b</span><span class="o">=</span><span class="p">[</span><span class="mf">.5</span><span class="p">,</span> <span class="mf">.5</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">x_a</span> <span class="o">=</span> <span class="p">[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">x_b</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">ot</span><span class="o">.</span><span class="n">emd2_1d</span><span class="p">(</span><span class="n">x_a</span><span class="p">,</span> <span class="n">x_b</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="go">0.5</span>
<span class="gp">>>> </span><span class="n">ot</span><span class="o">.</span><span class="n">emd2_1d</span><span class="p">(</span><span class="n">x_a</span><span class="p">,</span> <span class="n">x_b</span><span class="p">)</span>
<span class="go">0.5</span>
</pre></div>
</div>
<p class="rubric">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id8" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></span>
<p>Peyré, G., & Cuturi, M. (2017). “Computational Optimal
Transport”, 2018.</p>
</aside>
</aside>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="gen_modules/ot.lp.html#ot.lp.emd2" title="ot.lp.emd2"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ot.lp.emd2</span></code></a></dt><dd><p>EMD for multidimensional distributions</p>
</dd>
<dt><a class="reference internal" href="gen_modules/ot.lp.html#ot.lp.emd_1d" title="ot.lp.emd_1d"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ot.lp.emd_1d</span></code></a></dt><dd><p>EMD for 1d distributions (returns the transportation matrix instead of the cost)</p>
</dd>
</dl>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.emd_1d">
<span class="sig-prename descclassname"><span class="pre">ot.</span></span><span class="sig-name descname"><span class="pre">emd_1d</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x_a</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x_b</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">metric</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'sqeuclidean'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dense</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">check_marginals</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/ot/lp/solver_1d.html#emd_1d"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.emd_1d" title="Link to this definition"></a></dt>
<dd><p>Solves the Earth Movers distance problem between 1d measures and returns
the OT matrix</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\gamma = arg\min_\gamma \sum_i \sum_j \gamma_{ij} d(x_a[i], x_b[j])\\s.t. \gamma 1 = a,
\gamma^T 1= b,
\gamma\geq 0\end{aligned}\end{align} \]</div>
<p>where :</p>
<ul class="simple">
<li><p>d is the metric</p></li>
<li><p><span class="math notranslate nohighlight">\(x_a\)</span> and <span class="math notranslate nohighlight">\(x_b\)</span> are the samples</p></li>
<li><p>a and b are the sample weights</p></li>
</ul>
<p>This implementation only supports metrics
of the form <span class="math notranslate nohighlight">\(d(x, y) = |x - y|^p\)</span>.</p>
<p>Uses the algorithm detailed in <a href="#id86"><span class="problematic" id="id9">[1]_</span></a></p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>x_a</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>,</em><em>) or </em><em>(</em><em>ns</em><em>, </em><em>1</em><em>)</em>) – Source dirac locations (on the real line)</p></li>
<li><p><strong>x_b</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>nt</em><em>,</em><em>) or </em><em>(</em><em>ns</em><em>, </em><em>1</em><em>)</em>) – Target dirac locations (on the real line)</p></li>
<li><p><strong>a</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Source histogram (default is uniform weight)</p></li>
<li><p><strong>b</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>nt</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Target histogram (default is uniform weight)</p></li>
<li><p><strong>metric</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.13)"><em>str</em></a><em>, </em><em>optional</em><em> (</em><em>default='sqeuclidean'</em><em>)</em>) – Metric to be used. Only works with either of the strings
<cite>‘sqeuclidean’</cite>, <cite>‘minkowski’</cite>, <cite>‘cityblock’</cite>, or <cite>‘euclidean’</cite>.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em><em> (</em><em>default=1.0</em><em>)</em>) – The p-norm to apply for if metric=’minkowski’</p></li>
<li><p><strong>dense</strong> (<em>boolean</em><em>, </em><em>optional</em><em> (</em><em>default=True</em><em>)</em>) – If True, returns <span class="math notranslate nohighlight">\(\gamma\)</span> as a dense ndarray of shape (ns, nt).
Otherwise returns a sparse representation using scipy’s <cite>coo_matrix</cite>
format. Due to implementation details, this function runs faster when
<cite>‘sqeuclidean’</cite>, <cite>‘minkowski’</cite>, <cite>‘cityblock’</cite>, or <cite>‘euclidean’</cite> metrics
are used.</p></li>
<li><p><strong>log</strong> (<em>boolean</em><em>, </em><em>optional</em><em> (</em><em>default=False</em><em>)</em>) – If True, returns a dictionary containing the cost.
Otherwise returns only the optimal transportation matrix.</p></li>
<li><p><strong>check_marginals</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em><em> (</em><em>default=True</em><em>)</em>) – If True, checks that the marginals mass are equal. If False, skips the
check.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>gamma</strong> (<em>ndarray, shape (ns, nt)</em>) – Optimal transportation matrix for the given parameters</p></li>
<li><p><strong>log</strong> (<em>dict</em>) – If input log is True, a dictionary containing the cost</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<p>Simple example with obvious solution. The function emd_1d accepts lists and
performs automatic conversion to numpy arrays</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">ot</span>
<span class="gp">>>> </span><span class="n">a</span><span class="o">=</span><span class="p">[</span><span class="mf">.5</span><span class="p">,</span> <span class="mf">.5</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">b</span><span class="o">=</span><span class="p">[</span><span class="mf">.5</span><span class="p">,</span> <span class="mf">.5</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">x_a</span> <span class="o">=</span> <span class="p">[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">x_b</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">ot</span><span class="o">.</span><span class="n">emd_1d</span><span class="p">(</span><span class="n">x_a</span><span class="p">,</span> <span class="n">x_b</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="go">array([[0. , 0.5],</span>
<span class="go"> [0.5, 0. ]])</span>
<span class="gp">>>> </span><span class="n">ot</span><span class="o">.</span><span class="n">emd_1d</span><span class="p">(</span><span class="n">x_a</span><span class="p">,</span> <span class="n">x_b</span><span class="p">)</span>
<span class="go">array([[0. , 0.5],</span>
<span class="go"> [0.5, 0. ]])</span>
</pre></div>
</div>
<p class="rubric">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id10" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></span>
<p>Peyré, G., & Cuturi, M. (2017). “Computational Optimal
Transport”, 2018.</p>
</aside>
</aside>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="gen_modules/ot.lp.html#ot.lp.emd" title="ot.lp.emd"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ot.lp.emd</span></code></a></dt><dd><p>EMD for multidimensional distributions</p>
</dd>
<dt><a class="reference internal" href="gen_modules/ot.lp.html#ot.lp.emd2_1d" title="ot.lp.emd2_1d"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ot.lp.emd2_1d</span></code></a></dt><dd><p>EMD for 1d distributions (returns cost instead of the transportation matrix)</p>
</dd>
</dl>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.factored_optimal_transport">
<span class="sig-prename descclassname"><span class="pre">ot.</span></span><span class="sig-name descname"><span class="pre">factored_optimal_transport</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Xa</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Xb</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">reg</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">r</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">100</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">X0</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">stopThr</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-07</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">numItermax</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">100</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">verbose</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/ot/factored.html#factored_optimal_transport"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.factored_optimal_transport" title="Link to this definition"></a></dt>
<dd><p>Solves factored OT problem and return OT plans and intermediate distribution</p>
<p>This function solve the following OT problem <a href="#id87"><span class="problematic" id="id11">[40]_</span></a></p>
<div class="math notranslate nohighlight">
\[\mathop{\arg \min}_\mu \quad W_2^2(\mu_a,\mu)+ W_2^2(\mu,\mu_b)\]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\mu_a\)</span> and <span class="math notranslate nohighlight">\(\mu_b\)</span> are empirical distributions.</p></li>
<li><p><span class="math notranslate nohighlight">\(\mu\)</span> is an empirical distribution with r samples</p></li>
</ul>
<p>And returns the two OT plans between</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function is backend-compatible and will work on arrays
from all compatible backends. But the algorithm uses the C++ CPU backend
which can lead to copy overhead on GPU arrays.</p>
</div>
<p>Uses the conditional gradient algorithm to solve the problem proposed in
<a class="reference internal" href="#references-weak"><span class="std std-ref">[39]</span></a>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>Xa</strong> (<em>(</em><em>ns</em><em>,</em><em>d</em><em>) </em><em>array-like</em><em>, </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a>) – Source samples</p></li>
<li><p><strong>Xb</strong> (<em>(</em><em>nt</em><em>,</em><em>d</em><em>) </em><em>array-like</em><em>, </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a>) – Target samples</p></li>
<li><p><strong>a</strong> (<em>(</em><em>ns</em><em>,</em><em>) </em><em>array-like</em><em>, </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a>) – Source histogram (uniform weight if empty list)</p></li>
<li><p><strong>b</strong> (<em>(</em><em>nt</em><em>,</em><em>) </em><em>array-like</em><em>, </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a>) – Target histogram (uniform weight if empty list))</p></li>
<li><p><strong>numItermax</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.13)"><em>int</em></a><em>, </em><em>optional</em>) – Max number of iterations</p></li>
<li><p><strong>stopThr</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em>) – Stop threshold on the relative variation (>0)</p></li>
<li><p><strong>verbose</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em>) – Print information along iterations</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em>) – record log if True</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>Ga</strong> (<em>array-like, shape (ns, r)</em>) – Optimal transportation matrix between source and the intermediate
distribution</p></li>
<li><p><strong>Gb</strong> (<em>array-like, shape (r, nt)</em>) – Optimal transportation matrix between the intermediate and target
distribution</p></li>
<li><p><strong>X</strong> (<em>array-like, shape (r, d)</em>) – Support of the intermediate distribution</p></li>
<li><p><strong>log</strong> (<em>dict, optional</em>) – If input log is true, a dictionary containing the cost and dual
variables and exit status</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric" id="references-factored">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id12" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>40<span class="fn-bracket">]</span></span>
<p>Forrow, A., Hütter, J. C., Nitzan, M., Rigollet, P., Schiebinger,
G., & Weed, J. (2019, April). Statistical optimal transport via factored
couplings. In The 22nd International Conference on Artificial
Intelligence and Statistics (pp. 2454-2465). PMLR.</p>
</aside>
</aside>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="gen_modules/ot.bregman.html#ot.bregman.sinkhorn" title="ot.bregman.sinkhorn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ot.bregman.sinkhorn</span></code></a></dt><dd><p>Entropic regularized OT</p>
</dd>
<dt><a class="reference internal" href="gen_modules/ot.optim.html#id0" title="ot.optim.cg"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ot.optim.cg</span></code></a></dt><dd><p>General regularized OT</p>
</dd>
</dl>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.fused_gromov_wasserstein">
<span class="sig-prename descclassname"><span class="pre">ot.</span></span><span class="sig-name descname"><span class="pre">fused_gromov_wasserstein</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">M</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">C1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">C2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">q</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">loss_fun</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'square_loss'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">symmetric</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">alpha</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">armijo</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">G0</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_iter</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">10000.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">tol_rel</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-09</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">tol_abs</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-09</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/ot/gromov/_gw.html#fused_gromov_wasserstein"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.fused_gromov_wasserstein" title="Link to this definition"></a></dt>
<dd><p>Returns the Fused Gromov-Wasserstein transport between <span class="math notranslate nohighlight">\((\mathbf{C_1}, \mathbf{Y_1}, \mathbf{p})\)</span> and <span class="math notranslate nohighlight">\((\mathbf{C_2}, \mathbf{Y_2}, \mathbf{q})\)</span>
with pairwise distance matrix <span class="math notranslate nohighlight">\(\mathbf{M}\)</span> between node feature matrices <span class="math notranslate nohighlight">\(\mathbf{Y_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{Y_2}\)</span> (see <a class="reference internal" href="gen_modules/ot.gromov.html#references-fused-gromov-wasserstein"><span class="std std-ref">[24]</span></a>).</p>
<p>The function solves the following optimization problem using Conditional Gradient:</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\mathbf{T}^* \in\mathop{\arg\min}_\mathbf{T} \quad (1 - \alpha) \langle \mathbf{T}, \mathbf{M} \rangle_F +
\alpha \sum_{i,j,k,l} L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j} \mathbf{T}_{k,l}\\s.t. \ \mathbf{T} \mathbf{1} &= \mathbf{p}\\ \mathbf{T}^T \mathbf{1} &= \mathbf{q}\\ \mathbf{T} &\geq 0\end{aligned}\end{align} \]</div>
<p>Where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\mathbf{M}\)</span>: metric cost matrix between features across domains</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{C_1}\)</span>: Metric cost matrix in the source space</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{C_2}\)</span>: Metric cost matrix in the target space</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{p}\)</span>: distribution in the source space</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{q}\)</span>: distribution in the target space</p></li>
<li><p><cite>L</cite>: loss function to account for the misfit between the similarity and feature matrices</p></li>
<li><p><span class="math notranslate nohighlight">\(\alpha\)</span>: trade-off parameter</p></li>
</ul>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function is backend-compatible and will work on arrays
from all compatible backends. But the algorithm uses the C++ CPU backend
which can lead to copy overhead on GPU arrays.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>All computations in the conditional gradient solver are done with
numpy to limit memory overhead.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function will cast the computed transport plan to the data
type of the provided input <span class="math notranslate nohighlight">\(\mathbf{M}\)</span>. Casting to an integer
tensor might result in a loss of precision. If this behaviour is
unwanted, please make sure to provide a floating point input.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>M</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>, </em><em>nt</em><em>)</em>) – Metric cost matrix between features across domains</p></li>
<li><p><strong>C1</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>, </em><em>ns</em><em>)</em>) – Metric cost matrix representative of the structure in the source space</p></li>
<li><p><strong>C2</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>nt</em><em>, </em><em>nt</em><em>)</em>) – Metric cost matrix representative of the structure in the target space</p></li>
<li><p><strong>p</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Distribution in the source space.
If let to its default value None, uniform distribution is taken.</p></li>
<li><p><strong>q</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>nt</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Distribution in the target space.
If let to its default value None, uniform distribution is taken.</p></li>
<li><p><strong>loss_fun</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.13)"><em>str</em></a><em>, </em><em>optional</em>) – Loss function used for the solver</p></li>
<li><p><strong>symmetric</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em>) – Either C1 and C2 are to be assumed symmetric or not.
If let to its default None value, a symmetry test will be conducted.
Else if set to True (resp. False), C1 and C2 will be assumed symmetric (resp. asymmetric).</p></li>
<li><p><strong>alpha</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em>) – Trade-off parameter (0 < alpha < 1)</p></li>
<li><p><strong>armijo</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em>) – If True the step of the line-search is found via an armijo research. Else closed form is used.
If there are convergence issues use False.</p></li>
<li><p><strong>G0</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>,</em><em>nt</em><em>)</em><em>, </em><em>optional</em>) – If None the initial transport plan of the solver is pq^T.
Otherwise G0 must satisfy marginal constraints and will be used as initial transport of the solver.</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.13)"><em>bool</em></a><em>, </em><em>optional</em>) – record log if True</p></li>
<li><p><strong>max_iter</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.13)"><em>int</em></a><em>, </em><em>optional</em>) – Max number of iterations</p></li>
<li><p><strong>tol_rel</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em>) – Stop threshold on relative error (>0)</p></li>
<li><p><strong>tol_abs</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.13)"><em>float</em></a><em>, </em><em>optional</em>) – Stop threshold on absolute error (>0)</p></li>
<li><p><strong>**kwargs</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#dict" title="(in Python v3.13)"><em>dict</em></a>) – parameters can be directly passed to the ot.optim.cg solver</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>T</strong> (array-like, shape (<cite>ns</cite>, <cite>nt</cite>)) – Optimal transportation matrix for the given parameters.</p></li>
<li><p><strong>log</strong> (<em>dict</em>) – Log dictionary return only if log==True in parameters.</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric" id="references-fused-gromov-wasserstein">References</p>
<aside class="footnote-list brackets">