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<section id="module-ot.weak">
<span id="ot-weak"></span><h1>ot.weak<a class="headerlink" href="#module-ot.weak" title="Link to this heading"></a></h1>
<p>Weak optimal ransport solvers</p>
<section id="functions">
<h2>Functions<a class="headerlink" href="#functions" title="Link to this heading"></a></h2>
<dl class="py function">
<dt class="sig sig-object py" id="ot.weak.weak_optimal_transport">
<span class="sig-prename descclassname"><span class="pre">ot.weak.</span></span><span class="sig-name descname"><span class="pre">weak_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">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">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="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/weak.html#weak_optimal_transport"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.weak.weak_optimal_transport" title="Link to this definition"></a></dt>
<dd><p>Solves the weak optimal transport problem between two empirical distributions</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\gamma = \mathop{\arg \min}_\gamma \quad \sum_i \mathbf{a}_i \left(\mathbf{X^a}_i - \frac{1}{\mathbf{a}_i} \sum_j \gamma_{ij} \mathbf{X^b}_j \right)^2\\s.t. \ \gamma \mathbf{1} = \mathbf{a}\\ \gamma^T \mathbf{1} = \mathbf{b}\\ \gamma \geq 0\end{aligned}\end{align} \]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(X^a\)</span> and <span class="math notranslate nohighlight">\(X^b\)</span> are the sample matrices.</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>
<p>Uses the conditional gradient algorithm to solve the problem proposed
in <a class="reference internal" href="../all.html#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>G0</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>) – initial guess (default is indep joint density)</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>numItermaxEmd</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 for emd</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>stopThr2</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 absolute 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>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" id="references-weak">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id1" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>39<span class="fn-bracket">]</span></span>
<p>Gozlan, N., Roberto, C., Samson, P. M., & Tetali, P. (2017).
Kantorovich duality for general transport costs and applications.
Journal of Functional Analysis, 273(11), 3327-3405.</p>
</aside>
</aside>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="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="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>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="id0">
<span class="sig-prename descclassname"><span class="pre">ot.weak.</span></span><span class="sig-name descname"><span class="pre">weak_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">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">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="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/weak.html#weak_optimal_transport"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#id0" title="Link to this definition"></a></dt>
<dd><p>Solves the weak optimal transport problem between two empirical distributions</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\gamma = \mathop{\arg \min}_\gamma \quad \sum_i \mathbf{a}_i \left(\mathbf{X^a}_i - \frac{1}{\mathbf{a}_i} \sum_j \gamma_{ij} \mathbf{X^b}_j \right)^2\\s.t. \ \gamma \mathbf{1} = \mathbf{a}\\ \gamma^T \mathbf{1} = \mathbf{b}\\ \gamma \geq 0\end{aligned}\end{align} \]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(X^a\)</span> and <span class="math notranslate nohighlight">\(X^b\)</span> are the sample matrices.</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>
<p>Uses the conditional gradient algorithm to solve the problem proposed
in <a class="reference internal" href="../all.html#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>G0</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>) – initial guess (default is indep joint density)</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>numItermaxEmd</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 for emd</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>stopThr2</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 absolute 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>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" id="id2">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id3" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>39<span class="fn-bracket">]</span></span>
<p>Gozlan, N., Roberto, C., Samson, P. M., & Tetali, P. (2017).
Kantorovich duality for general transport costs and applications.
Journal of Functional Analysis, 273(11), 3327-3405.</p>
</aside>
</aside>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="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="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>
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