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<!DOCTYPE html>
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<li class="toctree-l3"><a class="reference internal" href="#id27"><code class="docutils literal notranslate"><span class="pre">smooth_ot_dual()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#id32"><code class="docutils literal notranslate"><span class="pre">smooth_ot_semi_dual()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#id37"><code class="docutils literal notranslate"><span class="pre">solve_dual()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#id38"><code class="docutils literal notranslate"><span class="pre">solve_semi_dual()</span></code></a></li>
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<section id="module-ot.smooth">
<span id="ot-smooth"></span><h1>ot.smooth<a class="headerlink" href="#module-ot.smooth" title="Link to this heading"></a></h1>
<p>Smooth and Sparse (KL an L2 reg.) and sparsity-constrained OT solvers.</p>
<p>Implementation of :
Smooth and Sparse Optimal Transport.
Mathieu Blondel, Vivien Seguy, Antoine Rolet.
In Proc. of AISTATS 2018.
<a class="reference external" href="https://arxiv.org/abs/1710.06276">https://arxiv.org/abs/1710.06276</a></p>
<p>(Original code from <a class="reference external" href="https://github.com/mblondel/smooth-ot/">https://github.com/mblondel/smooth-ot/</a>)</p>
<p>Sparsity-Constrained Optimal Transport.
Liu, T., Puigcerver, J., & Blondel, M. (2023).
Sparsity-constrained optimal transport.
Proceedings of the Eleventh International Conference on
Learning Representations (ICLR).
<a class="reference external" href="https://arxiv.org/abs/2209.15466">https://arxiv.org/abs/2209.15466</a></p>
<p>[17] Blondel, M., Seguy, V., & Rolet, A. (2018). Smooth and Sparse Optimal
Transport. Proceedings of the Twenty-First International Conference on
Artificial Intelligence and Statistics (AISTATS).</p>
<p>[50] Liu, T., Puigcerver, J., & Blondel, M. (2023).
Sparsity-constrained optimal transport.
Proceedings of the Eleventh International Conference on
Learning Representations (ICLR).</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.smooth.dual_obj_grad">
<span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">dual_obj_grad</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">alpha</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">beta</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">C</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">regul</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#dual_obj_grad"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.dual_obj_grad" title="Link to this definition"></a></dt>
<dd><p>Compute objective value and gradients of dual objective.</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>alpha</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><em>a</em><em>)</em>)</p></li>
<li><p><strong>beta</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em>) – Current iterate of dual potentials.</p></li>
<li><p><strong>a</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><em>a</em><em>)</em>)</p></li>
<li><p><strong>b</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em>) – Input histograms (should be non-negative and sum to 1).</p></li>
<li><p><strong>C</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Ground cost matrix.</p></li>
<li><p><strong>regul</strong> (<em>Regularization object</em>) – Should implement a <cite>delta_Omega(X)</cite> method.</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>obj</strong> (<em>float</em>) – Objective value (higher is better).</p></li>
<li><p><strong>grad_alpha</strong> (<em>array, shape = len(a)</em>) – Gradient w.r.t. <cite>alpha</cite>.</p></li>
<li><p><strong>grad_beta</strong> (<em>array, shape = len(b)</em>) – Gradient w.r.t. <cite>beta</cite>.</p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.smooth.get_plan_from_dual">
<span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">get_plan_from_dual</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">alpha</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">beta</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">C</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">regul</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#get_plan_from_dual"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.get_plan_from_dual" title="Link to this definition"></a></dt>
<dd><p>Retrieve optimal transportation plan from optimal dual potentials.</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>alpha</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><em>a</em><em>)</em>)</p></li>
<li><p><strong>beta</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em>) – Optimal dual potentials.</p></li>
<li><p><strong>C</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Ground cost matrix.</p></li>
<li><p><strong>regul</strong> (<em>Regularization object</em>) – Should implement a <cite>delta_Omega(X)</cite> method.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>T</strong> – Optimal transportation plan.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>array, shape = (len(a), len(<a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b">b</a>))</p>
</dd>
</dl>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.smooth.get_plan_from_semi_dual">
<span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">get_plan_from_semi_dual</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">alpha</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">C</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">regul</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#get_plan_from_semi_dual"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.get_plan_from_semi_dual" title="Link to this definition"></a></dt>
<dd><p>Retrieve optimal transportation plan from optimal semi-dual potentials.</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>alpha</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><em>a</em><em>)</em>) – Optimal semi-dual potentials.</p></li>
<li><p><strong>b</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em>) – Second input histogram (should be non-negative and sum to 1).</p></li>
<li><p><strong>C</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Ground cost matrix.</p></li>
<li><p><strong>regul</strong> (<em>Regularization object</em>) – Should implement a <cite>delta_Omega(X)</cite> method.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>T</strong> – Optimal transportation plan.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>array, shape = (len(a), len(<a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b">b</a>))</p>
</dd>
</dl>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.smooth.projection_simplex">
<span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">projection_simplex</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">V</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z</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">axis</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/smooth.html#projection_simplex"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.projection_simplex" title="Link to this definition"></a></dt>
<dd><p>Projection of <span class="math notranslate nohighlight">\(\mathbf{V}\)</span> onto the simplex, scaled by <cite>z</cite></p>
<div class="math notranslate nohighlight">
\[\begin{split}P\left(\mathbf{V}, z\right) = \mathop{\arg \min}_{\substack{\mathbf{y} >= 0 \\ \sum_i \mathbf{y}_i = z}} \quad \|\mathbf{y} - \mathbf{V}\|^2\end{split}\]</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>V</strong> (<em>ndarray</em><em>, </em><em>rank 2</em>)</p></li>
<li><p><strong>z</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> or </em><em>array</em>) – If array, len(z) must be compatible with <span class="math notranslate nohighlight">\(\mathbf{V}\)</span></p></li>
<li><p><strong>axis</strong> (<em>None</em><em> or </em><a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a>) – <ul>
<li><p>axis=None: project <span class="math notranslate nohighlight">\(\mathbf{V}\)</span> by <span class="math notranslate nohighlight">\(P(\mathbf{V}.\mathrm{ravel}(), z)\)</span></p></li>
<li><p>axis=1: project each <span class="math notranslate nohighlight">\(\mathbf{V}_i\)</span> by <span class="math notranslate nohighlight">\(P(\mathbf{V}_i, z_i)\)</span></p></li>
<li><p>axis=0: project each <span class="math notranslate nohighlight">\(\mathbf{V}_{:, j}\)</span> by <span class="math notranslate nohighlight">\(P(\mathbf{V}_{:, j}, z_j)\)</span></p></li>
</ul>
</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>projection</strong></p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray, shape <span class="math notranslate nohighlight">\(\mathbf{V}\)</span>.shape</p>
</dd>
</dl>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.smooth.semi_dual_obj_grad">
<span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">semi_dual_obj_grad</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">alpha</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">C</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">regul</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#semi_dual_obj_grad"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.semi_dual_obj_grad" title="Link to this definition"></a></dt>
<dd><p>Compute objective value and gradient of semi-dual objective.</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>alpha</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><em>a</em><em>)</em>) – Current iterate of semi-dual potentials.</p></li>
<li><p><strong>a</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><em>a</em><em>)</em>)</p></li>
<li><p><strong>b</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em>) – Input histograms (should be non-negative and sum to 1).</p></li>
<li><p><strong>C</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Ground cost matrix.</p></li>
<li><p><strong>regul</strong> (<em>Regularization object</em>) – Should implement a <cite>max_Omega(X)</cite> method.</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>obj</strong> (<em>float</em>) – Objective value (higher is better).</p></li>
<li><p><strong>grad</strong> (<em>array, shape = len(a)</em>) – Gradient w.r.t. alpha.</p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.smooth.smooth_ot_dual">
<span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">smooth_ot_dual</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">reg</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">reg_type</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'l2'</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">'L-BFGS-B'</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-09</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">500</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">max_nz</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/smooth.html#smooth_ot_dual"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.smooth_ot_dual" title="Link to this definition"></a></dt>
<dd><p>Solve the regularized OT problem in the dual and return the OT matrix</p>
<p>The function solves the smooth relaxed dual formulation (7) in
<span class="xref std std-ref">[17]</span>:</p>
<div class="math notranslate nohighlight">
\[\max_{\alpha,\beta}\quad \mathbf{a}^T\alpha + \mathbf{b}^T\beta -
\sum_j \delta_\Omega \left(\alpha+\beta_j-\mathbf{m}_j \right)\]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\mathbf{m}_j\)</span> is the j-th column of the cost matrix</p></li>
<li><p><span class="math notranslate nohighlight">\(\delta_\Omega\)</span> is the convex conjugate of the regularization term <span class="math notranslate nohighlight">\(\Omega\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{a}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> are source and target weights (sum to 1)</p></li>
</ul>
<p>The OT matrix can is reconstructed from the gradient of <span class="math notranslate nohighlight">\(\delta_\Omega\)</span>
(See <span class="xref std std-ref">[17]</span> Proposition 1).
The optimization algorithm is using gradient decent (L-BFGS by default).</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>np.ndarray</em><em> (</em><em>ns</em><em>,</em><em>)</em>) – samples weights in the source domain</p></li>
<li><p><strong>b</strong> (<em>np.ndarray</em><em> (</em><em>nt</em><em>,</em><em>) or </em><em>np.ndarray</em><em> (</em><em>nt</em><em>,</em><em>nbb</em><em>)</em>) – samples in the target domain, compute sinkhorn with multiple targets
and fixed <span class="math notranslate nohighlight">\(\mathbf{M}\)</span> if <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> is a matrix
(return OT loss + dual variables in log)</p></li>
<li><p><strong>M</strong> (<em>np.ndarray</em><em> (</em><em>ns</em><em>,</em><em>nt</em><em>)</em>) – loss matrix</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>) – Regularization term >0</p></li>
<li><p><strong>reg_type</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)"><em>str</em></a>) – <p>Regularization type, can be the following (default =’l2’):</p>
<blockquote>
<div><ul>
<li><p>’kl’ : Kullback Leibler (~ Neg-entropy used in sinkhorn
<span class="xref std std-ref">[2]</span>)</p></li>
<li><p>’l2’ : Squared Euclidean regularization</p></li>
<li><p>’sparsity_constrained’ : Sparsity-constrained regularization [50]</p></li>
</ul>
</div></blockquote>
</p></li>
<li><p><strong>max_nz</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> or </em><em>None</em><em>, </em><em>optional. Used only in the case</em><em> of </em><em>reg_type = 'sparsity_constrained' to specify the maximum number</em><em> of </em><em>nonzeros per column</em><em> of </em><em>the optimal plan;</em>) – not used for other regularization types.</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>) – Solver to use for scipy.optimize.minimize</p></li>
<li><p><strong>numItermax</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>) – Max number of iterations</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#bool" title="(in Python v3.15)"><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/builtins/functions.html#bool" title="(in Python v3.15)"><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>(ns, nt) ndarray</em>) – 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-smooth-ot-dual">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id1" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></span>
<p>M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport,
Advances in Neural Information Processing Systems (NIPS) 26, 2013</p>
</aside>
<aside class="footnote brackets" id="id2" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>17<span class="fn-bracket">]</span></span>
<p>Blondel, M., Seguy, V., & Rolet, A. (2018). Smooth and Sparse Optimal Transport.
Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics (AISTATS).</p>
</aside>
<aside class="footnote brackets" id="id3" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>50<span class="fn-bracket">]</span></span>
<p>Liu, T., Puigcerver, J., & Blondel, M. (2023). Sparsity-constrained optimal transport.
Proceedings of the Eleventh International Conference on Learning Representations (ICLR).</p>
</aside>
</aside>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="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>Unregularized OT</p>
</dd>
<dt><a class="reference internal" href="../all.html#ot.sinkhorn" title="ot.sinkhorn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ot.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>
<section id="examples-using-ot-smooth-smooth-ot-dual">
<h3>Examples using <code class="docutils literal notranslate"><span class="pre">ot.smooth.smooth_ot_dual</span></code><a class="headerlink" href="#examples-using-ot-smooth-smooth-ot-dual" title="Link to this heading"></a></h3>
<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="This example illustrates the computation of EMD and Sinkhorn transport plans and their visualization."><img alt="" src="../_images/sphx_glr_plot_OT_1D_thumb.png" />
<p><a class="reference internal" href="../auto_examples/plot_OT_1D.html"><span class="doc">Optimal Transport for fixed support</span></a></p>
<div class="sphx-glr-thumbnail-title">Optimal Transport for fixed support</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="Quickstart guide to the POT toolbox."><img alt="" src="../_images/sphx_glr_plot_quickstart_guide_thumb.png" />
<p><a class="reference internal" href="../auto_examples/plot_quickstart_guide.html"><span class="doc">Quickstart Guide</span></a></p>
<div class="sphx-glr-thumbnail-title">Quickstart Guide</div>
</div></div><div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.smooth.smooth_ot_semi_dual">
<span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">smooth_ot_semi_dual</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">reg</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">reg_type</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'l2'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_nz</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">'L-BFGS-B'</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-09</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">500</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><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#smooth_ot_semi_dual"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.smooth_ot_semi_dual" title="Link to this definition"></a></dt>
<dd><p>Solve the regularized OT problem in the semi-dual and return the OT matrix</p>
<p>The function solves the smooth relaxed dual formulation (10) in
<span class="xref std std-ref">[17]</span>:</p>
<div class="math notranslate nohighlight">
\[\max_{\alpha}\quad \mathbf{a}^T\alpha- \mathrm{OT}_\Omega^*(\alpha, \mathbf{b})\]</div>
<p>where :</p>
<div class="math notranslate nohighlight">
\[\mathrm{OT}_\Omega^*(\alpha,b)=\sum_j \mathbf{b}_j\]</div>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\mathbf{m}_j\)</span> is the j-th column of the cost matrix</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathrm{OT}_\Omega^*(\alpha,b)\)</span> is defined in Eq. (9) in
<span class="xref std std-ref">[17]</span></p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{a}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> are source and target weights (sum to 1)</p></li>
</ul>
<p>The OT matrix can is reconstructed using <span class="xref std std-ref">[17]</span> Proposition 2.
The optimization algorithm is using gradient decent (L-BFGS by default).</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>np.ndarray</em><em> (</em><em>ns</em><em>,</em><em>)</em>) – samples weights in the source domain</p></li>
<li><p><strong>b</strong> (<em>np.ndarray</em><em> (</em><em>nt</em><em>,</em><em>) or </em><em>np.ndarray</em><em> (</em><em>nt</em><em>,</em><em>nbb</em><em>)</em>) – samples in the target domain, compute sinkhorn with multiple targets
and fixed:math:<cite>mathbf{M}</cite> if <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> is a matrix
(return OT loss + dual variables in log)</p></li>
<li><p><strong>M</strong> (<em>np.ndarray</em><em> (</em><em>ns</em><em>,</em><em>nt</em><em>)</em>) – loss matrix</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>) – Regularization term >0</p></li>
<li><p><strong>reg_type</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)"><em>str</em></a>) – <p>Regularization type, can be the following (default =’l2’):</p>
<blockquote>
<div><ul>
<li><p>’kl’ : Kullback Leibler (~ Neg-entropy used in sinkhorn
<span class="xref std std-ref">[2]</span>)</p></li>
<li><p>’l2’ : Squared Euclidean regularization</p></li>
<li><p>’sparsity_constrained’ : Sparsity-constrained regularization [50]</p></li>
</ul>
</div></blockquote>
</p></li>
<li><p><strong>max_nz</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> or </em><em>None</em><em>, </em><em>optional. Used only in the case</em><em> of </em><em>reg_type = 'sparsity_constrained' to specify the maximum number</em><em> of </em><em>nonzeros per column</em><em> of </em><em>the optimal plan;</em>) – not used for other regularization types.</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>) – Solver to use for scipy.optimize.minimize</p></li>
<li><p><strong>numItermax</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>) – Max number of iterations</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#bool" title="(in Python v3.15)"><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/builtins/functions.html#bool" title="(in Python v3.15)"><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>(ns, nt) ndarray</em>) – 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-smooth-ot-semi-dual">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id4" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></span>
<p>M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport,
Advances in Neural Information Processing Systems (NIPS) 26, 2013</p>
</aside>
<aside class="footnote brackets" id="id5" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>17<span class="fn-bracket">]</span></span>
<p>Blondel, M., Seguy, V., & Rolet, A. (2018). Smooth and Sparse Optimal Transport.
Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics (AISTATS).</p>
</aside>
<aside class="footnote brackets" id="id6" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>50<span class="fn-bracket">]</span></span>
<p>Liu, T., Puigcerver, J., & Blondel, M. (2023). Sparsity-constrained optimal transport.
Proceedings of the Eleventh International Conference on Learning Representations (ICLR).</p>
</aside>
</aside>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="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>Unregularized OT</p>
</dd>
<dt><a class="reference internal" href="../all.html#ot.sinkhorn" title="ot.sinkhorn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ot.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="ot.smooth.solve_dual">
<span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">solve_dual</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">C</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">regul</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">'L-BFGS-B'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">tol</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">max_iter</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">500</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><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#solve_dual"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.solve_dual" title="Link to this definition"></a></dt>
<dd><p>Solve the “smoothed” dual objective.</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</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>)</em>)</p></li>
<li><p><strong>b</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>, </em><em>)</em>) – Input histograms (should be non-negative and sum to 1).</p></li>
<li><p><strong>C</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Ground cost matrix.</p></li>
<li><p><strong>regul</strong> (<em>Regularization object</em>) – Should implement a <cite>delta_Omega(X)</cite> method.</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>) – Solver to be used (passed to <cite>scipy.optimize.minimize</cite>).</p></li>
<li><p><strong>tol</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a>) – Tolerance parameter.</p></li>
<li><p><strong>max_iter</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a>) – Maximum number of 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>alpha</strong> (<em>array, shape = (len(a), )</em>)</p></li>
<li><p><strong>beta</strong> (<em>array, shape = (len(b), )</em>) – Dual potentials.</p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.smooth.solve_semi_dual">
<span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">solve_semi_dual</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">C</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">regul</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">'L-BFGS-B'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">tol</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">max_iter</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">500</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><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#solve_semi_dual"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.solve_semi_dual" title="Link to this definition"></a></dt>
<dd><p>Solve the “smoothed” semi-dual objective.</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</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>)</em>)</p></li>
<li><p><strong>b</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>, </em><em>)</em>) – Input histograms (should be non-negative and sum to 1).</p></li>
<li><p><strong>C</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Ground cost matrix.</p></li>
<li><p><strong>regul</strong> (<em>Regularization object</em>) – Should implement a <cite>max_Omega(X)</cite> method.</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>) – Solver to be used (passed to <cite>scipy.optimize.minimize</cite>).</p></li>
<li><p><strong>tol</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)"><em>float</em></a>) – Tolerance parameter.</p></li>
<li><p><strong>max_iter</strong> (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)"><em>int</em></a>) – Maximum number of iterations.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>alpha</strong> – Semi-dual potentials.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>array, shape = (len(a), )</p>
</dd>
</dl>
</dd></dl>
<div class="sphx-glr-clear"></div></section>
</section>
<section id="classes">
<h2>Classes<a class="headerlink" href="#classes" title="Link to this heading"></a></h2>
<dl class="py class">
<dt class="sig sig-object py" id="ot.smooth.NegEntropy">
<span class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></span><span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">NegEntropy</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">gamma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#NegEntropy"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.NegEntropy" title="Link to this definition"></a></dt>
<dd><p>NegEntropy regularization</p>
<dl class="py method">
<dt class="sig sig-object py" id="ot.smooth.NegEntropy.Omega">
<span class="sig-name descname"><span class="pre">Omega</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">T</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#NegEntropy.Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.NegEntropy.Omega" title="Link to this definition"></a></dt>
<dd><p>Compute regularization term.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>T</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><em>a</em><em>) </em><em>x len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em>) – Input array.</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>value</strong> – Regularization term.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="ot.smooth.NegEntropy.delta_Omega">
<span class="sig-name descname"><span class="pre">delta_Omega</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#NegEntropy.delta_Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.NegEntropy.delta_Omega" title="Link to this definition"></a></dt>
<dd><p>Compute <span class="math notranslate nohighlight">\(\delta_\Omega(\mathbf{X}_{:, j})\)</span> for each <span class="math notranslate nohighlight">\(\mathbf{X}_{:, j}\)</span>.</p>
<div class="math notranslate nohighlight">
\[\delta_\Omega(\mathbf{x}) = \sup_{\mathbf{y} >= 0} \
\mathbf{y}^T \mathbf{x} - \Omega(\mathbf{y})\]</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>X</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Input array.</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>v</strong> (<em>array, (len(b), )</em>) – Values: <span class="math notranslate nohighlight">\(\mathbf{v}_j = \delta_\Omega(\mathbf{X}_{:, j})\)</span></p></li>
<li><p><strong>G</strong> (<em>array, (len(a), len(b))</em>) – Gradients: <span class="math notranslate nohighlight">\(\mathbf{G}_{:, j} = \nabla \delta_\Omega(\mathbf{X}_{:, j})\)</span></p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="ot.smooth.NegEntropy.max_Omega">
<span class="sig-name descname"><span class="pre">max_Omega</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">b</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#NegEntropy.max_Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.NegEntropy.max_Omega" title="Link to this definition"></a></dt>
<dd><p>Compute <span class="math notranslate nohighlight">\(\mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span> for each <span class="math notranslate nohighlight">\(\mathbf{X}_{:, j}\)</span>.</p>
<div class="math notranslate nohighlight">
\[\mathrm{max}_{\Omega, j}(\mathbf{x}) =
\sup_{\substack{\mathbf{y} >= 0 \ \sum_i \mathbf{y}_i = 1}}
\mathbf{y}^T \mathbf{x} - \frac{1}{\mathbf{b}_j} \Omega(\mathbf{b}_j \mathbf{y})\]</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>X</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Input array.</p></li>
<li><p><strong>b</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>, </em><em>)</em>)</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>v</strong> (<em>array, (len(b), )</em>) – Values: <span class="math notranslate nohighlight">\(\mathbf{v}_j = \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span></p></li>
<li><p><strong>G</strong> (<em>array, (len(a), len(b))</em>) – Gradients: <span class="math notranslate nohighlight">\(\mathbf{G}_{:, j} = \nabla \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span></p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py class">
<dt class="sig sig-object py" id="ot.smooth.Regularization">
<span class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></span><span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">Regularization</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">gamma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#Regularization"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.Regularization" title="Link to this definition"></a></dt>
<dd><p>Base class for Regularization objects</p>
<p class="rubric">Notes</p>
<p>This class is not intended for direct use but as apparent for true
regularization implementation.</p>
<dl class="py method">
<dt class="sig sig-object py" id="ot.smooth.Regularization.Omega">
<span class="sig-name descname"><span class="pre">Omega</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#Regularization.Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.Regularization.Omega" title="Link to this definition"></a></dt>
<dd><p>Compute regularization term.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>T</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><em>a</em><em>) </em><em>x len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em>) – Input array.</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>value</strong> – Regularization term.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="ot.smooth.Regularization.delta_Omega">
<span class="sig-name descname"><span class="pre">delta_Omega</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#Regularization.delta_Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.Regularization.delta_Omega" title="Link to this definition"></a></dt>
<dd><p>Compute <span class="math notranslate nohighlight">\(\delta_\Omega(\mathbf{X}_{:, j})\)</span> for each <span class="math notranslate nohighlight">\(\mathbf{X}_{:, j}\)</span>.</p>
<div class="math notranslate nohighlight">
\[\delta_\Omega(\mathbf{x}) = \sup_{\mathbf{y} >= 0} \
\mathbf{y}^T \mathbf{x} - \Omega(\mathbf{y})\]</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>X</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Input array.</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>v</strong> (<em>array, (len(b), )</em>) – Values: <span class="math notranslate nohighlight">\(\mathbf{v}_j = \delta_\Omega(\mathbf{X}_{:, j})\)</span></p></li>
<li><p><strong>G</strong> (<em>array, (len(a), len(b))</em>) – Gradients: <span class="math notranslate nohighlight">\(\mathbf{G}_{:, j} = \nabla \delta_\Omega(\mathbf{X}_{:, j})\)</span></p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="ot.smooth.Regularization.max_Omega">
<span class="sig-name descname"><span class="pre">max_Omega</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">b</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#Regularization.max_Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.Regularization.max_Omega" title="Link to this definition"></a></dt>
<dd><p>Compute <span class="math notranslate nohighlight">\(\mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span> for each <span class="math notranslate nohighlight">\(\mathbf{X}_{:, j}\)</span>.</p>
<div class="math notranslate nohighlight">
\[\mathrm{max}_{\Omega, j}(\mathbf{x}) =
\sup_{\substack{\mathbf{y} >= 0 \ \sum_i \mathbf{y}_i = 1}}
\mathbf{y}^T \mathbf{x} - \frac{1}{\mathbf{b}_j} \Omega(\mathbf{b}_j \mathbf{y})\]</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>X</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Input array.</p></li>
<li><p><strong>b</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>, </em><em>)</em>)</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>v</strong> (<em>array, (len(b), )</em>) – Values: <span class="math notranslate nohighlight">\(\mathbf{v}_j = \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span></p></li>
<li><p><strong>G</strong> (<em>array, (len(a), len(b))</em>) – Gradients: <span class="math notranslate nohighlight">\(\mathbf{G}_{:, j} = \nabla \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span></p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py class">
<dt class="sig sig-object py" id="ot.smooth.SparsityConstrained">
<span class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></span><span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">SparsityConstrained</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">max_nz</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">gamma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#SparsityConstrained"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.SparsityConstrained" title="Link to this definition"></a></dt>
<dd><p>Squared L2 regularization with sparsity constraints</p>
<dl class="py method">
<dt class="sig sig-object py" id="ot.smooth.SparsityConstrained.Omega">
<span class="sig-name descname"><span class="pre">Omega</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">T</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#SparsityConstrained.Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.SparsityConstrained.Omega" title="Link to this definition"></a></dt>
<dd><p>Compute regularization term.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>T</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><em>a</em><em>) </em><em>x len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em>) – Input array.</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>value</strong> – Regularization term.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="ot.smooth.SparsityConstrained.delta_Omega">
<span class="sig-name descname"><span class="pre">delta_Omega</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#SparsityConstrained.delta_Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.SparsityConstrained.delta_Omega" title="Link to this definition"></a></dt>
<dd><p>Compute <span class="math notranslate nohighlight">\(\delta_\Omega(\mathbf{X}_{:, j})\)</span> for each <span class="math notranslate nohighlight">\(\mathbf{X}_{:, j}\)</span>.</p>
<div class="math notranslate nohighlight">
\[\delta_\Omega(\mathbf{x}) = \sup_{\mathbf{y} >= 0} \
\mathbf{y}^T \mathbf{x} - \Omega(\mathbf{y})\]</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>X</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Input array.</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>v</strong> (<em>array, (len(b), )</em>) – Values: <span class="math notranslate nohighlight">\(\mathbf{v}_j = \delta_\Omega(\mathbf{X}_{:, j})\)</span></p></li>
<li><p><strong>G</strong> (<em>array, (len(a), len(b))</em>) – Gradients: <span class="math notranslate nohighlight">\(\mathbf{G}_{:, j} = \nabla \delta_\Omega(\mathbf{X}_{:, j})\)</span></p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="ot.smooth.SparsityConstrained.max_Omega">
<span class="sig-name descname"><span class="pre">max_Omega</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">b</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#SparsityConstrained.max_Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.SparsityConstrained.max_Omega" title="Link to this definition"></a></dt>
<dd><p>Compute <span class="math notranslate nohighlight">\(\mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span> for each <span class="math notranslate nohighlight">\(\mathbf{X}_{:, j}\)</span>.</p>
<div class="math notranslate nohighlight">
\[\mathrm{max}_{\Omega, j}(\mathbf{x}) =
\sup_{\substack{\mathbf{y} >= 0 \ \sum_i \mathbf{y}_i = 1}}
\mathbf{y}^T \mathbf{x} - \frac{1}{\mathbf{b}_j} \Omega(\mathbf{b}_j \mathbf{y})\]</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>X</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Input array.</p></li>
<li><p><strong>b</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>, </em><em>)</em>)</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>v</strong> (<em>array, (len(b), )</em>) – Values: <span class="math notranslate nohighlight">\(\mathbf{v}_j = \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span></p></li>
<li><p><strong>G</strong> (<em>array, (len(a), len(b))</em>) – Gradients: <span class="math notranslate nohighlight">\(\mathbf{G}_{:, j} = \nabla \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span></p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py class">
<dt class="sig sig-object py" id="ot.smooth.SquaredL2">
<span class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></span><span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">SquaredL2</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">gamma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#SquaredL2"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.SquaredL2" title="Link to this definition"></a></dt>
<dd><p>Squared L2 regularization</p>
<dl class="py method">
<dt class="sig sig-object py" id="ot.smooth.SquaredL2.Omega">
<span class="sig-name descname"><span class="pre">Omega</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">T</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#SquaredL2.Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.SquaredL2.Omega" title="Link to this definition"></a></dt>
<dd><p>Compute regularization term.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>T</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><em>a</em><em>) </em><em>x len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em>) – Input array.</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>value</strong> – Regularization term.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="ot.smooth.SquaredL2.delta_Omega">
<span class="sig-name descname"><span class="pre">delta_Omega</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#SquaredL2.delta_Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.SquaredL2.delta_Omega" title="Link to this definition"></a></dt>
<dd><p>Compute <span class="math notranslate nohighlight">\(\delta_\Omega(\mathbf{X}_{:, j})\)</span> for each <span class="math notranslate nohighlight">\(\mathbf{X}_{:, j}\)</span>.</p>
<div class="math notranslate nohighlight">
\[\delta_\Omega(\mathbf{x}) = \sup_{\mathbf{y} >= 0} \
\mathbf{y}^T \mathbf{x} - \Omega(\mathbf{y})\]</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>X</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Input array.</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>v</strong> (<em>array, (len(b), )</em>) – Values: <span class="math notranslate nohighlight">\(\mathbf{v}_j = \delta_\Omega(\mathbf{X}_{:, j})\)</span></p></li>
<li><p><strong>G</strong> (<em>array, (len(a), len(b))</em>) – Gradients: <span class="math notranslate nohighlight">\(\mathbf{G}_{:, j} = \nabla \delta_\Omega(\mathbf{X}_{:, j})\)</span></p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="ot.smooth.SquaredL2.max_Omega">
<span class="sig-name descname"><span class="pre">max_Omega</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">b</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#SquaredL2.max_Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.smooth.SquaredL2.max_Omega" title="Link to this definition"></a></dt>
<dd><p>Compute <span class="math notranslate nohighlight">\(\mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span> for each <span class="math notranslate nohighlight">\(\mathbf{X}_{:, j}\)</span>.</p>
<div class="math notranslate nohighlight">
\[\mathrm{max}_{\Omega, j}(\mathbf{x}) =
\sup_{\substack{\mathbf{y} >= 0 \ \sum_i \mathbf{y}_i = 1}}
\mathbf{y}^T \mathbf{x} - \frac{1}{\mathbf{b}_j} \Omega(\mathbf{b}_j \mathbf{y})\]</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>X</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Input array.</p></li>
<li><p><strong>b</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>, </em><em>)</em>)</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>v</strong> (<em>array, (len(b), )</em>) – Values: <span class="math notranslate nohighlight">\(\mathbf{v}_j = \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span></p></li>
<li><p><strong>G</strong> (<em>array, (len(a), len(b))</em>) – Gradients: <span class="math notranslate nohighlight">\(\mathbf{G}_{:, j} = \nabla \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span></p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
</dd></dl>
<div class="sphx-glr-clear"></div></section>
<dl class="py class">
<dt class="sig sig-object py" id="id0">
<span class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></span><span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">NegEntropy</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">gamma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#NegEntropy"><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>NegEntropy regularization</p>
<dl class="py method">
<dt class="sig sig-object py" id="id7">
<span class="sig-name descname"><span class="pre">Omega</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">T</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#NegEntropy.Omega"><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>Compute regularization term.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>T</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><em>a</em><em>) </em><em>x len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em>) – Input array.</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>value</strong> – Regularization term.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="id8">
<span class="sig-name descname"><span class="pre">delta_Omega</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#NegEntropy.delta_Omega"><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>Compute <span class="math notranslate nohighlight">\(\delta_\Omega(\mathbf{X}_{:, j})\)</span> for each <span class="math notranslate nohighlight">\(\mathbf{X}_{:, j}\)</span>.</p>
<div class="math notranslate nohighlight">
\[\delta_\Omega(\mathbf{x}) = \sup_{\mathbf{y} >= 0} \
\mathbf{y}^T \mathbf{x} - \Omega(\mathbf{y})\]</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>X</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Input array.</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>v</strong> (<em>array, (len(b), )</em>) – Values: <span class="math notranslate nohighlight">\(\mathbf{v}_j = \delta_\Omega(\mathbf{X}_{:, j})\)</span></p></li>
<li><p><strong>G</strong> (<em>array, (len(a), len(b))</em>) – Gradients: <span class="math notranslate nohighlight">\(\mathbf{G}_{:, j} = \nabla \delta_\Omega(\mathbf{X}_{:, j})\)</span></p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="id9">
<span class="sig-name descname"><span class="pre">max_Omega</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">b</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#NegEntropy.max_Omega"><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>Compute <span class="math notranslate nohighlight">\(\mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span> for each <span class="math notranslate nohighlight">\(\mathbf{X}_{:, j}\)</span>.</p>
<div class="math notranslate nohighlight">
\[\mathrm{max}_{\Omega, j}(\mathbf{x}) =
\sup_{\substack{\mathbf{y} >= 0 \ \sum_i \mathbf{y}_i = 1}}
\mathbf{y}^T \mathbf{x} - \frac{1}{\mathbf{b}_j} \Omega(\mathbf{b}_j \mathbf{y})\]</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>X</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Input array.</p></li>
<li><p><strong>b</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>, </em><em>)</em>)</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>v</strong> (<em>array, (len(b), )</em>) – Values: <span class="math notranslate nohighlight">\(\mathbf{v}_j = \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span></p></li>
<li><p><strong>G</strong> (<em>array, (len(a), len(b))</em>) – Gradients: <span class="math notranslate nohighlight">\(\mathbf{G}_{:, j} = \nabla \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span></p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
</dd></dl>
<dl class="py class">
<dt class="sig sig-object py" id="id10">
<span class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></span><span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">Regularization</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">gamma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#Regularization"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#id10" title="Link to this definition"></a></dt>
<dd><p>Base class for Regularization objects</p>
<p class="rubric">Notes</p>
<p>This class is not intended for direct use but as apparent for true
regularization implementation.</p>
<dl class="py method">
<dt class="sig sig-object py" id="id11">
<span class="sig-name descname"><span class="pre">Omega</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#Regularization.Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#id11" title="Link to this definition"></a></dt>
<dd><p>Compute regularization term.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>T</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><em>a</em><em>) </em><em>x len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em>) – Input array.</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>value</strong> – Regularization term.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="id12">
<span class="sig-name descname"><span class="pre">delta_Omega</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#Regularization.delta_Omega"><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>Compute <span class="math notranslate nohighlight">\(\delta_\Omega(\mathbf{X}_{:, j})\)</span> for each <span class="math notranslate nohighlight">\(\mathbf{X}_{:, j}\)</span>.</p>
<div class="math notranslate nohighlight">
\[\delta_\Omega(\mathbf{x}) = \sup_{\mathbf{y} >= 0} \
\mathbf{y}^T \mathbf{x} - \Omega(\mathbf{y})\]</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>X</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Input array.</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>v</strong> (<em>array, (len(b), )</em>) – Values: <span class="math notranslate nohighlight">\(\mathbf{v}_j = \delta_\Omega(\mathbf{X}_{:, j})\)</span></p></li>
<li><p><strong>G</strong> (<em>array, (len(a), len(b))</em>) – Gradients: <span class="math notranslate nohighlight">\(\mathbf{G}_{:, j} = \nabla \delta_\Omega(\mathbf{X}_{:, j})\)</span></p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="id13">
<span class="sig-name descname"><span class="pre">max_Omega</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">b</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#Regularization.max_Omega"><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>Compute <span class="math notranslate nohighlight">\(\mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span> for each <span class="math notranslate nohighlight">\(\mathbf{X}_{:, j}\)</span>.</p>
<div class="math notranslate nohighlight">
\[\mathrm{max}_{\Omega, j}(\mathbf{x}) =
\sup_{\substack{\mathbf{y} >= 0 \ \sum_i \mathbf{y}_i = 1}}
\mathbf{y}^T \mathbf{x} - \frac{1}{\mathbf{b}_j} \Omega(\mathbf{b}_j \mathbf{y})\]</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>X</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Input array.</p></li>
<li><p><strong>b</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>, </em><em>)</em>)</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>v</strong> (<em>array, (len(b), )</em>) – Values: <span class="math notranslate nohighlight">\(\mathbf{v}_j = \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span></p></li>
<li><p><strong>G</strong> (<em>array, (len(a), len(b))</em>) – Gradients: <span class="math notranslate nohighlight">\(\mathbf{G}_{:, j} = \nabla \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})\)</span></p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
</dd></dl>
<dl class="py class">
<dt class="sig sig-object py" id="id14">
<span class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></span><span class="sig-prename descclassname"><span class="pre">ot.smooth.</span></span><span class="sig-name descname"><span class="pre">SparsityConstrained</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">max_nz</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">gamma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#SparsityConstrained"><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>Squared L2 regularization with sparsity constraints</p>
<dl class="py method">
<dt class="sig sig-object py" id="id15">
<span class="sig-name descname"><span class="pre">Omega</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">T</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#SparsityConstrained.Omega"><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>Compute regularization term.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>T</strong> (<em>array</em><em>, </em><em>shape = len</em><em>(</em><em>a</em><em>) </em><em>x len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em>) – Input array.</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>value</strong> – Regularization term.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a></p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="id16">
<span class="sig-name descname"><span class="pre">delta_Omega</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/smooth.html#SparsityConstrained.delta_Omega"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#id16" title="Link to this definition"></a></dt>
<dd><p>Compute <span class="math notranslate nohighlight">\(\delta_\Omega(\mathbf{X}_{:, j})\)</span> for each <span class="math notranslate nohighlight">\(\mathbf{X}_{:, j}\)</span>.</p>
<div class="math notranslate nohighlight">
\[\delta_\Omega(\mathbf{x}) = \sup_{\mathbf{y} >= 0} \
\mathbf{y}^T \mathbf{x} - \Omega(\mathbf{y})\]</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>X</strong> (<em>array</em><em>, </em><em>shape =</em><em> (</em><em>len</em><em>(</em><em>a</em><em>)</em><em>, </em><em>len</em><em>(</em><a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b"><em>b</em></a><em>)</em><em>)</em>) – Input array.</p>