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<section id="module-ot.dr">
<span id="ot-dr"></span><h1>ot.dr<a class="headerlink" href="#module-ot.dr" title="Link to this heading"></a></h1>
<p>Dimension reduction with OT</p>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>Note that by default the module is not imported in <a class="reference internal" href="../all.html#module-ot" title="ot"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot</span></code></a>. In order to
use it you need to explicitly import <a class="reference internal" href="#module-ot.dr" title="ot.dr"><code class="xref py py-mod docutils literal notranslate"><span class="pre">ot.dr</span></code></a></p>
</div>
<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.dr.dist">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</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></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#dist"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.dr.dist" title="Link to this definition"></a></dt>
<dd><p>Compute squared euclidean distance between samples (autograd)</p>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.dr.ewca">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">ewca</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">U0</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">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</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">method</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'BCD'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sinkhorn_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">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">maxiter</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">maxiter_sink</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">maxiter_MM</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">verbose</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#ewca"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.dr.ewca" title="Link to this definition"></a></dt>
<dd><p>Entropic Wasserstein Component Analysis <span class="xref std std-ref">[52]</span>.</p>
<p>The function solves the following optimization problem:</p>
<div class="math notranslate nohighlight">
\[\mathbf{U} = \mathop{\arg \min}_\mathbf{U} \quad
W(\mathbf{X}, \mathbf{U}\mathbf{U}^T \mathbf{X})\]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\mathbf{U}\)</span> is a matrix in the Stiefel(<cite>p</cite>, <cite>d</cite>) manifold</p></li>
<li><p><span class="math notranslate nohighlight">\(W\)</span> is entropic regularized Wasserstein distances</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{X}\)</span> are samples</p></li>
</ul>
<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</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>d</em><em>)</em>) – Samples from measure <span class="math notranslate nohighlight">\(\mu\)</span>.</p></li>
<li><p><strong>U0</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>d</em><em>, </em><em>k</em><em>)</em><em>, </em><em>optional</em>) – Initial starting point for projection.</p></li>
<li><p><strong>reg</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Regularization term >0 (entropic regularization).</p></li>
<li><p><strong>k</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Subspace dimension.</p></li>
<li><p><strong>method</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)"><em>str</em></a><em>, </em><em>optional</em>) – Either ‘BCD’ or ‘MM’ (Block Coordinate Descent or Majorization-Minimization).
Prefer MM when d is large.</p></li>
<li><p><strong>sinkhorn_method</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)"><em>str</em></a>) – Method used for the Sinkhorn solver, see <span class="xref std std-ref">ot.bregman.sinkhorn</span> for more details.</p></li>
<li><p><strong>stopThr</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Stop threshold on error (>0).</p></li>
<li><p><strong>maxiter</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Maximum number of iterations of the BCD/MM.</p></li>
<li><p><strong>maxiter_sink</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Maximum number of iterations of the Sinkhorn solver.</p></li>
<li><p><strong>maxiter_MM</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Maximum number of iterations of the MM (only used when method=’MM’).</p></li>
<li><p><strong>verbose</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Print information along iterations.</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>pi</strong> (<em>ndarray, shape (n, n)</em>) – Optimal transportation matrix for the given parameters.</p></li>
<li><p><strong>U</strong> (<em>ndarray, shape (d, k)</em>) – Matrix Stiefel manifold.</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric" id="references-entropic-wasserstein-component-analysis">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id1" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>52<span class="fn-bracket">]</span></span>
<p>Collas, A., Vayer, T., Flamary, F., & Breloy, A. (2023).
Entropic Wasserstein Component Analysis.</p>
</aside>
</aside>
</dd></dl>
<section id="examples-using-ot-dr-ewca">
<h3>Examples using <code class="docutils literal notranslate"><span class="pre">ot.dr.ewca</span></code><a class="headerlink" href="#examples-using-ot-dr-ewca" title="Link to this heading"></a></h3>
<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="This example illustrates the use of EWCA as proposed in [52]."><img alt="" src="../_images/sphx_glr_plot_EWCA_thumb.png" />
<p><a class="reference internal" href="../auto_examples/others/plot_EWCA.html"><span class="doc">Entropic Wasserstein Component Analysis</span></a></p>
<div class="sphx-glr-thumbnail-title">Entropic Wasserstein Component Analysis</div>
</div></div><div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.dr.fda">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">fda</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</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">reg</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-16</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#fda"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.dr.fda" title="Link to this definition"></a></dt>
<dd><p>Fisher Discriminant Analysis</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</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>d</em><em>)</em>) – Training samples.</p></li>
<li><p><strong>y</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>,</em><em>)</em>) – Labels for training samples.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Size of dimensionality reduction.</p></li>
<li><p><strong>reg</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Regularization term >0 (ridge regularization)</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>P</strong> (<em>ndarray, shape (d, p)</em>) – Optimal transportation matrix for the given parameters</p></li>
<li><p><strong>proj</strong> (<em>callable</em>) – projection function including mean centering</p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
</section>
<section id="examples-using-ot-dr-fda">
<h3>Examples using <code class="docutils literal notranslate"><span class="pre">ot.dr.fda</span></code><a class="headerlink" href="#examples-using-ot-dr-fda" title="Link to this heading"></a></h3>
<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="This example illustrate the use of WDA as proposed in [11]."><img alt="" src="../_images/sphx_glr_plot_WDA_thumb.png" />
<p><a class="reference internal" href="../auto_examples/others/plot_WDA.html"><span class="doc">Wasserstein Discriminant Analysis</span></a></p>
<div class="sphx-glr-thumbnail-title">Wasserstein Discriminant Analysis</div>
</div></div><div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.dr.logsumexp">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">logsumexp</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">axis</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#logsumexp"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.dr.logsumexp" title="Link to this definition"></a></dt>
<dd><p>Log-sum-exp reduction compatible with autograd (no numpy implementation)</p>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.dr.projection_robust_wasserstein">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">projection_robust_wasserstein</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Y</span></span></em>, <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">tau</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">U0</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">0.1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</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">stopThr</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.001</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">maxiter</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">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">random_state</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/dr.html#projection_robust_wasserstein"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.dr.projection_robust_wasserstein" title="Link to this definition"></a></dt>
<dd><p>Projection Robust Wasserstein Distance <span class="xref std std-ref">[32]</span></p>
<p>The function solves the following optimization problem:</p>
<div class="math notranslate nohighlight">
\[\max_{U \in St(d, k)} \ \min_{\pi \in \Pi(\mu,\nu)} \quad \sum_{i,j} \pi_{i,j}
\|U^T(\mathbf{x}_i - \mathbf{y}_j)\|^2 - \mathrm{reg} \cdot H(\pi)\]</div>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(U\)</span> is a linear projection operator in the Stiefel(<cite>d</cite>, <cite>k</cite>) manifold</p></li>
<li><p><span class="math notranslate nohighlight">\(H(\pi)\)</span> is entropy regularizer</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{x}_i\)</span>, <span class="math notranslate nohighlight">\(\mathbf{y}_j\)</span> are samples of measures <span class="math notranslate nohighlight">\(\mu\)</span> and <span class="math notranslate nohighlight">\(\nu\)</span> respectively</p></li>
</ul>
<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</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>d</em><em>)</em>) – Samples from measure <span class="math notranslate nohighlight">\(\mu\)</span></p></li>
<li><p><strong>Y</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>d</em><em>)</em>) – Samples from measure <span class="math notranslate nohighlight">\(\nu\)</span></p></li>
<li><p><strong>a</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>)</em>) – weights for measure <span class="math notranslate nohighlight">\(\mu\)</span></p></li>
<li><p><strong>b</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>)</em>) – weights for measure <span class="math notranslate nohighlight">\(\nu\)</span></p></li>
<li><p><strong>tau</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a>) – stepsize for Riemannian Gradient Descent</p></li>
<li><p><strong>U0</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>d</em><em>, </em><em>p</em><em>)</em>) – Initial starting point for projection.</p></li>
<li><p><strong>reg</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Regularization term >0 (entropic regularization)</p></li>
<li><p><strong>k</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a>) – Subspace dimension</p></li>
<li><p><strong>stopThr</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><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/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Print information along iterations.</p></li>
<li><p><strong>random_state</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>RandomState instance</em><em> or </em><em>None</em><em>, </em><em>default=None</em>) – Determines random number generation for initial value of projection
operator when U0 is not given.</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>pi</strong> (<em>ndarray, shape (n, n)</em>) – Optimal transportation matrix for the given parameters</p></li>
<li><p><strong>U</strong> (<em>ndarray, shape (d, k)</em>) – Projection operator.</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric" id="references-projection-robust-wasserstein">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id2" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>32<span class="fn-bracket">]</span></span>
<p>Huang, M. , Ma S. & Lai L. (2021).
A Riemannian Block Coordinate Descent Method for Computing
the Projection Robust Wasserstein Distance, ICML.</p>
</aside>
</aside>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.dr.sinkhorn">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">sinkhorn</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">w1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">w2</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">k</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#sinkhorn"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.dr.sinkhorn" title="Link to this definition"></a></dt>
<dd><p>Sinkhorn algorithm with fixed number of iteration (autograd)</p>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.dr.sinkhorn_log">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">sinkhorn_log</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">w1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">w2</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">k</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#sinkhorn_log"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.dr.sinkhorn_log" title="Link to this definition"></a></dt>
<dd><p>Sinkhorn algorithm in log-domain with fixed number of iteration (autograd)</p>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.dr.split_classes">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">split_classes</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#split_classes"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.dr.split_classes" title="Link to this definition"></a></dt>
<dd><p>split samples in <span class="math notranslate nohighlight">\(\mathbf{X}\)</span> by classes in <span class="math notranslate nohighlight">\(\mathbf{y}\)</span></p>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.dr.wda">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">wda</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</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">reg</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">k</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">solver</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">sinkhorn_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">maxiter</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">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">P0</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">normalize</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">random_state</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">step_growth</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">step_shrink</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">max_backtracks</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">40</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">gtol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-12</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#wda"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.dr.wda" title="Link to this definition"></a></dt>
<dd><p>Wasserstein Discriminant Analysis <span class="xref std std-ref">[11]</span></p>
<p>The function solves the following optimization problem:</p>
<div class="math notranslate nohighlight">
\[\mathbf{P} = \mathop{\arg \min}_\mathbf{P} \quad
\frac{\sum\limits_i W(P \mathbf{X}^i, P \mathbf{X}^i)}{\sum\limits_{i, j \neq i} W(P \mathbf{X}^i, P \mathbf{X}^j)}\]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(P\)</span> is a linear projection operator in the Stiefel(<cite>p</cite>, <cite>d</cite>) manifold</p></li>
<li><p><span class="math notranslate nohighlight">\(W\)</span> is entropic regularized Wasserstein distances</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{X}^i\)</span> are samples in the dataset corresponding to class i</p></li>
</ul>
<p><strong>Choosing a Sinkhorn solver</strong></p>
<p>By default and when using a regularization parameter that is not too small
the default sinkhorn solver should be enough. If you need to use a small
regularization to get sparse cost matrices, you should use the
<a class="reference internal" href="#id13" title="ot.dr.sinkhorn_log"><code class="xref py py-func docutils literal notranslate"><span class="pre">ot.dr.sinkhorn_log()</span></code></a> solver that will avoid numerical
errors, but can be slow in practice.</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</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>d</em><em>)</em>) – Training samples.</p></li>
<li><p><strong>y</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>,</em><em>)</em>) – Labels for training samples.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Size of dimensionality reduction.</p></li>
<li><p><strong>reg</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Regularization term >0 (entropic regularization)</p></li>
<li><p><strong>solver</strong> (<em>None</em><em> | </em><a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)"><em>str</em></a><em> | </em><em>pymanopt.optimizers</em><em>, </em><em>optional</em>) – <p>Chooses both the autodiff framework and the optimizer.</p>
<ul>
<li><p><cite>None</cite> or <cite>‘autograd’</cite> (default): autograd and pymanopt
<cite>SteepestDescent</cite>.</p></li>
<li><p><cite>’TrustRegions’</cite> (or <cite>‘tr’</cite>): autograd and pymanopt <cite>TrustRegions</cite>.</p></li>
<li><p>a <cite>pymanopt.optimizers</cite> instance: autograd with that optimizer.</p></li>
<li><p><cite>’torch’</cite>: PyTorch autodiff with Riemannian gradient descent and a QR
retraction. Requires only <cite>torch</cite>, so it works on installations
without autograd or pymanopt, and accepts torch tensors directly,
keeping their device and dtype.</p></li>
</ul>
</p></li>
<li><p><strong>sinkhorn_method</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)"><em>str</em></a>) – method used for the Sinkhorn solver, either ‘sinkhorn’ or ‘sinkhorn_log’</p></li>
<li><p><strong>P0</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>d</em><em>, </em><em>p</em><em>)</em>) – Initial starting point for projection.</p></li>
<li><p><strong>normalize</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#bool" title="(in Python v3.15)"><em>bool</em></a><em>, </em><em>optional</em>) – Normalize the Wasserstaiun distance by the average distance on P0 (default : False)</p></li>
<li><p><strong>verbose</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Print information along iterations.</p></li>
<li><p><strong>random_state</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>RandomState instance</em><em> or </em><em>None</em><em>, </em><em>optional</em>) – Seeds the random starting point when <cite>P0</cite> is None. Only used by
<cite>solver=’torch’</cite>; the pymanopt solvers draw their own starting point.</p></li>
<li><p><strong>step_growth</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Factor applied to the last accepted step size to start the next line
search (default 2.0). Only used by <cite>solver=’torch’</cite>.</p></li>
<li><p><strong>step_shrink</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Factor applied to the step size on each backtracking step
(default 0.5). Only used by <cite>solver=’torch’</cite>.</p></li>
<li><p><strong>max_backtracks</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Maximum number of backtracking steps per iteration before the solver
stops (default 40). Only used by <cite>solver=’torch’</cite>.</p></li>
<li><p><strong>gtol</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Stop once the norm of the tangent gradient falls below this value
(default 1e-12). Only used by <cite>solver=’torch’</cite>.</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>P</strong> (<em>ndarray, shape (d, p)</em>) – Optimal transportation matrix for the given parameters</p></li>
<li><p><strong>proj</strong> (<em>callable</em>) – Projection function including mean centering.</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric" id="references-wda">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id3" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>11<span class="fn-bracket">]</span></span>
<p>Flamary, R., Cuturi, M., Courty, N., & Rakotomamonjy, A. (2016).
Wasserstein Discriminant Analysis. arXiv preprint arXiv:1608.08063.</p>
</aside>
</aside>
</dd></dl>
</section>
<section id="examples-using-ot-dr-wda">
<h3>Examples using <code class="docutils literal notranslate"><span class="pre">ot.dr.wda</span></code><a class="headerlink" href="#examples-using-ot-dr-wda" title="Link to this heading"></a></h3>
<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="This example illustrate the use of WDA as proposed in [11]."><img alt="" src="../_images/sphx_glr_plot_WDA_thumb.png" />
<p><a class="reference internal" href="../auto_examples/others/plot_WDA.html"><span class="doc">Wasserstein Discriminant Analysis</span></a></p>
<div class="sphx-glr-thumbnail-title">Wasserstein Discriminant Analysis</div>
</div></div><div class="sphx-glr-clear"></div></section>
</section>
<dl class="py function">
<dt class="sig sig-object py" id="id0">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</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></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#dist"><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>Compute squared euclidean distance between samples (autograd)</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="id4">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">ewca</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">U0</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">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</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">method</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'BCD'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sinkhorn_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">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">maxiter</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">maxiter_sink</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">maxiter_MM</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">verbose</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#ewca"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#id4" title="Link to this definition"></a></dt>
<dd><p>Entropic Wasserstein Component Analysis <span class="xref std std-ref">[52]</span>.</p>
<p>The function solves the following optimization problem:</p>
<div class="math notranslate nohighlight">
\[\mathbf{U} = \mathop{\arg \min}_\mathbf{U} \quad
W(\mathbf{X}, \mathbf{U}\mathbf{U}^T \mathbf{X})\]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\mathbf{U}\)</span> is a matrix in the Stiefel(<cite>p</cite>, <cite>d</cite>) manifold</p></li>
<li><p><span class="math notranslate nohighlight">\(W\)</span> is entropic regularized Wasserstein distances</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{X}\)</span> are samples</p></li>
</ul>
<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</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>d</em><em>)</em>) – Samples from measure <span class="math notranslate nohighlight">\(\mu\)</span>.</p></li>
<li><p><strong>U0</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>d</em><em>, </em><em>k</em><em>)</em><em>, </em><em>optional</em>) – Initial starting point for projection.</p></li>
<li><p><strong>reg</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Regularization term >0 (entropic regularization).</p></li>
<li><p><strong>k</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Subspace dimension.</p></li>
<li><p><strong>method</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)"><em>str</em></a><em>, </em><em>optional</em>) – Either ‘BCD’ or ‘MM’ (Block Coordinate Descent or Majorization-Minimization).
Prefer MM when d is large.</p></li>
<li><p><strong>sinkhorn_method</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)"><em>str</em></a>) – Method used for the Sinkhorn solver, see <span class="xref std std-ref">ot.bregman.sinkhorn</span> for more details.</p></li>
<li><p><strong>stopThr</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Stop threshold on error (>0).</p></li>
<li><p><strong>maxiter</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Maximum number of iterations of the BCD/MM.</p></li>
<li><p><strong>maxiter_sink</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Maximum number of iterations of the Sinkhorn solver.</p></li>
<li><p><strong>maxiter_MM</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Maximum number of iterations of the MM (only used when method=’MM’).</p></li>
<li><p><strong>verbose</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Print information along iterations.</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>pi</strong> (<em>ndarray, shape (n, n)</em>) – Optimal transportation matrix for the given parameters.</p></li>
<li><p><strong>U</strong> (<em>ndarray, shape (d, k)</em>) – Matrix Stiefel manifold.</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric" id="id5">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id6" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>52<span class="fn-bracket">]</span></span>
<p>Collas, A., Vayer, T., Flamary, F., & Breloy, A. (2023).
Entropic Wasserstein Component Analysis.</p>
</aside>
</aside>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="id7">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">fda</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</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">reg</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-16</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#fda"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#id7" title="Link to this definition"></a></dt>
<dd><p>Fisher Discriminant Analysis</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</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>d</em><em>)</em>) – Training samples.</p></li>
<li><p><strong>y</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>,</em><em>)</em>) – Labels for training samples.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Size of dimensionality reduction.</p></li>
<li><p><strong>reg</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Regularization term >0 (ridge regularization)</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>P</strong> (<em>ndarray, shape (d, p)</em>) – Optimal transportation matrix for the given parameters</p></li>
<li><p><strong>proj</strong> (<em>callable</em>) – projection function including mean centering</p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="id8">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">logsumexp</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">axis</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#logsumexp"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#id8" title="Link to this definition"></a></dt>
<dd><p>Log-sum-exp reduction compatible with autograd (no numpy implementation)</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="id9">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">projection_robust_wasserstein</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Y</span></span></em>, <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">tau</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">U0</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">0.1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</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">stopThr</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.001</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">maxiter</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">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">random_state</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/dr.html#projection_robust_wasserstein"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#id9" title="Link to this definition"></a></dt>
<dd><p>Projection Robust Wasserstein Distance <span class="xref std std-ref">[32]</span></p>
<p>The function solves the following optimization problem:</p>
<div class="math notranslate nohighlight">
\[\max_{U \in St(d, k)} \ \min_{\pi \in \Pi(\mu,\nu)} \quad \sum_{i,j} \pi_{i,j}
\|U^T(\mathbf{x}_i - \mathbf{y}_j)\|^2 - \mathrm{reg} \cdot H(\pi)\]</div>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(U\)</span> is a linear projection operator in the Stiefel(<cite>d</cite>, <cite>k</cite>) manifold</p></li>
<li><p><span class="math notranslate nohighlight">\(H(\pi)\)</span> is entropy regularizer</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{x}_i\)</span>, <span class="math notranslate nohighlight">\(\mathbf{y}_j\)</span> are samples of measures <span class="math notranslate nohighlight">\(\mu\)</span> and <span class="math notranslate nohighlight">\(\nu\)</span> respectively</p></li>
</ul>
<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</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>d</em><em>)</em>) – Samples from measure <span class="math notranslate nohighlight">\(\mu\)</span></p></li>
<li><p><strong>Y</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>d</em><em>)</em>) – Samples from measure <span class="math notranslate nohighlight">\(\nu\)</span></p></li>
<li><p><strong>a</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>)</em>) – weights for measure <span class="math notranslate nohighlight">\(\mu\)</span></p></li>
<li><p><strong>b</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>)</em>) – weights for measure <span class="math notranslate nohighlight">\(\nu\)</span></p></li>
<li><p><strong>tau</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a>) – stepsize for Riemannian Gradient Descent</p></li>
<li><p><strong>U0</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>d</em><em>, </em><em>p</em><em>)</em>) – Initial starting point for projection.</p></li>
<li><p><strong>reg</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Regularization term >0 (entropic regularization)</p></li>
<li><p><strong>k</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a>) – Subspace dimension</p></li>
<li><p><strong>stopThr</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><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/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Print information along iterations.</p></li>
<li><p><strong>random_state</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>RandomState instance</em><em> or </em><em>None</em><em>, </em><em>default=None</em>) – Determines random number generation for initial value of projection
operator when U0 is not given.</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>pi</strong> (<em>ndarray, shape (n, n)</em>) – Optimal transportation matrix for the given parameters</p></li>
<li><p><strong>U</strong> (<em>ndarray, shape (d, k)</em>) – Projection operator.</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric" id="id10">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id11" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>32<span class="fn-bracket">]</span></span>
<p>Huang, M. , Ma S. & Lai L. (2021).
A Riemannian Block Coordinate Descent Method for Computing
the Projection Robust Wasserstein Distance, ICML.</p>
</aside>
</aside>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="id12">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">sinkhorn</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">w1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">w2</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">k</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#sinkhorn"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#id12" title="Link to this definition"></a></dt>
<dd><p>Sinkhorn algorithm with fixed number of iteration (autograd)</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="id13">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">sinkhorn_log</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">w1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">w2</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">k</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#sinkhorn_log"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#id13" title="Link to this definition"></a></dt>
<dd><p>Sinkhorn algorithm in log-domain with fixed number of iteration (autograd)</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="id14">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">split_classes</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#split_classes"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#id14" title="Link to this definition"></a></dt>
<dd><p>split samples in <span class="math notranslate nohighlight">\(\mathbf{X}\)</span> by classes in <span class="math notranslate nohighlight">\(\mathbf{y}\)</span></p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="id15">
<span class="sig-prename descclassname"><span class="pre">ot.dr.</span></span><span class="sig-name descname"><span class="pre">wda</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</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">reg</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">k</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">solver</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">sinkhorn_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">maxiter</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">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">P0</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">normalize</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">random_state</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">step_growth</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">step_shrink</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">max_backtracks</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">40</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">gtol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-12</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/dr.html#wda"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#id15" title="Link to this definition"></a></dt>
<dd><p>Wasserstein Discriminant Analysis <span class="xref std std-ref">[11]</span></p>
<p>The function solves the following optimization problem:</p>
<div class="math notranslate nohighlight">
\[\mathbf{P} = \mathop{\arg \min}_\mathbf{P} \quad
\frac{\sum\limits_i W(P \mathbf{X}^i, P \mathbf{X}^i)}{\sum\limits_{i, j \neq i} W(P \mathbf{X}^i, P \mathbf{X}^j)}\]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(P\)</span> is a linear projection operator in the Stiefel(<cite>p</cite>, <cite>d</cite>) manifold</p></li>
<li><p><span class="math notranslate nohighlight">\(W\)</span> is entropic regularized Wasserstein distances</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{X}^i\)</span> are samples in the dataset corresponding to class i</p></li>
</ul>
<p><strong>Choosing a Sinkhorn solver</strong></p>
<p>By default and when using a regularization parameter that is not too small
the default sinkhorn solver should be enough. If you need to use a small
regularization to get sparse cost matrices, you should use the
<a class="reference internal" href="#id13" title="ot.dr.sinkhorn_log"><code class="xref py py-func docutils literal notranslate"><span class="pre">ot.dr.sinkhorn_log()</span></code></a> solver that will avoid numerical
errors, but can be slow in practice.</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</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>, </em><em>d</em><em>)</em>) – Training samples.</p></li>
<li><p><strong>y</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n</em><em>,</em><em>)</em>) – Labels for training samples.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Size of dimensionality reduction.</p></li>
<li><p><strong>reg</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Regularization term >0 (entropic regularization)</p></li>
<li><p><strong>solver</strong> (<em>None</em><em> | </em><a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)"><em>str</em></a><em> | </em><em>pymanopt.optimizers</em><em>, </em><em>optional</em>) – <p>Chooses both the autodiff framework and the optimizer.</p>
<ul>
<li><p><cite>None</cite> or <cite>‘autograd’</cite> (default): autograd and pymanopt
<cite>SteepestDescent</cite>.</p></li>
<li><p><cite>’TrustRegions’</cite> (or <cite>‘tr’</cite>): autograd and pymanopt <cite>TrustRegions</cite>.</p></li>
<li><p>a <cite>pymanopt.optimizers</cite> instance: autograd with that optimizer.</p></li>
<li><p><cite>’torch’</cite>: PyTorch autodiff with Riemannian gradient descent and a QR
retraction. Requires only <cite>torch</cite>, so it works on installations
without autograd or pymanopt, and accepts torch tensors directly,
keeping their device and dtype.</p></li>
</ul>
</p></li>
<li><p><strong>sinkhorn_method</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)"><em>str</em></a>) – method used for the Sinkhorn solver, either ‘sinkhorn’ or ‘sinkhorn_log’</p></li>
<li><p><strong>P0</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>d</em><em>, </em><em>p</em><em>)</em>) – Initial starting point for projection.</p></li>
<li><p><strong>normalize</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#bool" title="(in Python v3.15)"><em>bool</em></a><em>, </em><em>optional</em>) – Normalize the Wasserstaiun distance by the average distance on P0 (default : False)</p></li>
<li><p><strong>verbose</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Print information along iterations.</p></li>
<li><p><strong>random_state</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>RandomState instance</em><em> or </em><em>None</em><em>, </em><em>optional</em>) – Seeds the random starting point when <cite>P0</cite> is None. Only used by
<cite>solver=’torch’</cite>; the pymanopt solvers draw their own starting point.</p></li>
<li><p><strong>step_growth</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Factor applied to the last accepted step size to start the next line
search (default 2.0). Only used by <cite>solver=’torch’</cite>.</p></li>
<li><p><strong>step_shrink</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Factor applied to the step size on each backtracking step
(default 0.5). Only used by <cite>solver=’torch’</cite>.</p></li>
<li><p><strong>max_backtracks</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a><em>, </em><em>optional</em>) – Maximum number of backtracking steps per iteration before the solver
stops (default 40). Only used by <cite>solver=’torch’</cite>.</p></li>
<li><p><strong>gtol</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a><em>, </em><em>optional</em>) – Stop once the norm of the tangent gradient falls below this value
(default 1e-12). Only used by <cite>solver=’torch’</cite>.</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>P</strong> (<em>ndarray, shape (d, p)</em>) – Optimal transportation matrix for the given parameters</p></li>
<li><p><strong>proj</strong> (<em>callable</em>) – Projection function including mean centering.</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric" id="id16">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id17" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>11<span class="fn-bracket">]</span></span>
<p>Flamary, R., Cuturi, M., Courty, N., & Rakotomamonjy, A. (2016).
Wasserstein Discriminant Analysis. arXiv preprint arXiv:1608.08063.</p>
</aside>
</aside>
</dd></dl>
</section>
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