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<section id="module-ot.sgot">
<span id="ot-sgot"></span><h1>ot.sgot<a class="headerlink" href="#module-ot.sgot" title="Link to this heading"></a></h1>
<p>Spectral-Grassmann optimal transport for linear operators.</p>
<p>This module implements the Spectral-Grassmann Wasserstein framework for
comparing dynamical systems via their learned operator representations.</p>
<p>It provides tools to extract spectral “atoms” (eigenvalues and associated
eigenspaces) from linear operators and to compute an optimal transport metric
that combines a spectral term on eigenvalues with a Grassmannian term on
eigenspaces.</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.sgot.eigenvalue_cost_matrix">
<span class="sig-prename descclassname"><span class="pre">ot.sgot.</span></span><span class="sig-name descname"><span class="pre">eigenvalue_cost_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Ds</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Dt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">q</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eigen_scaling</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">nx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/sgot.html#eigenvalue_cost_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sgot.eigenvalue_cost_matrix" title="Link to this definition"></a></dt>
<dd><p>Compute pairwise eigenvalue distances for source and target domains.</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>Ds</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_s</em><em>,</em><em>)</em>) – Source eigenvalues.</p></li>
<li><p><strong>Dt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_t</em><em>,</em><em>)</em>) – Target eigenvalues.</p></li>
<li><p><strong>q</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em> or </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Exponent applied to the eigenvalue distance, default 1.</p></li>
<li><p><strong>eigen_scaling</strong> (<em>None</em><em> or </em><em>array-like</em><em> of </em><em>length 2</em><em>, </em><em>optional</em>) – Scaling (real_scale, imag_scale) applied to eigenvalues before computing
distances. If None, defaults to (1.0, 1.0). Accepts tuple/list or
array/tensor with two entries.</p></li>
<li><p><strong>nx</strong> (<em>module</em><em>, </em><em>optional</em>) – Backend (NumPy-compatible). If None, inferred from inputs.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>C</strong> – Eigenvalue cost matrix.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>np.ndarray, shape (n_s, n_t)</p>
</dd>
</dl>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.sgot.sgot_cost_matrix">
<span class="sig-prename descclassname"><span class="pre">ot.sgot.</span></span><span class="sig-name descname"><span class="pre">sgot_cost_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Ds</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Rs</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Ls</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Dt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Rt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Lt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eta</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">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">q</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">grassmann_metric</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'chordal'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eigen_scaling</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">nx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eps</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/sgot.html#sgot_cost_matrix"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sgot.sgot_cost_matrix" title="Link to this definition"></a></dt>
<dd><p>Compute the SGOT cost matrix between two spectral decompositions.</p>
<p>This returns the discrete ground cost matrix used in the SGOT optimal transport
objective. Each spectral atom is <span class="math notranslate nohighlight">\(z_i=(\lambda_i, V_i)\)</span> where
<span class="math notranslate nohighlight">\(\lambda_i \in \mathbb{C}\)</span> is an eigenvalue and <span class="math notranslate nohighlight">\(V_i\)</span> is the
associated (bi-orthogonal) eigenspace point.</p>
<div class="math notranslate nohighlight">
\[C_2(i,j) \;=\; \eta\,C_\lambda(i,j) \;+\; (1-\eta)\,C_G(i,j),\]</div>
<p>with spectral term</p>
<div class="math notranslate nohighlight">
\[C_\lambda(i,j) \;=\; \big|\lambda_i - \lambda'_j\big|^{q},\]</div>
<p>and Grassmann term computed from a similarity score <span class="math notranslate nohighlight">\(\delta_{ij}\in[0,1]\)</span>
built from left/right eigenvectors</p>
<div class="math notranslate nohighlight">
\[\delta_{ij} \;=\; \left|\langle r_i, r'_j\rangle\,\langle \ell_i, \ell'_j\rangle\right|.\]</div>
<p>Depending on <code class="docutils literal notranslate"><span class="pre">grassmann_metric</span></code>, the Grassmann contribution is:</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">"chordal"</span></code>: <span class="math notranslate nohighlight">\(C_G(i,j) = 1 - \delta_{ij}^2\)</span></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"geodesic"</span></code>: <span class="math notranslate nohighlight">\(C_G(i,j) = \arccos(\delta_{ij})^2\)</span></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"procrustes"</span></code>: <span class="math notranslate nohighlight">\(C_G(i,j) = 2(1-\delta_{ij})\)</span></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"martin"</span></code>: <span class="math notranslate nohighlight">\(C_G(i,j) = -\log\!\left(\max(\delta_{ij}^2,\varepsilon)\right)\)</span></p></li>
</ul>
<p>Finally, we return a matrix suited for a <span class="math notranslate nohighlight">\(p\)</span>-Wasserstein objective by
treating <span class="math notranslate nohighlight">\(C_2 \approx d^2\)</span> and outputting</p>
<div class="math notranslate nohighlight">
\[C(i,j) \;=\; \big(\operatorname{Re}(C_2(i,j))\big)^{p/2}.\]</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>Ds</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_s</em><em>,</em><em>)</em>) – Eigenvalues of operator T1.</p></li>
<li><p><strong>Rs</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_s</em><em>)</em>) – Right eigenvectors of operator T1.</p></li>
<li><p><strong>Ls</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_s</em><em>)</em>) – Left eigenvectors of operator T1.</p></li>
<li><p><strong>Dt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_t</em><em>,</em><em>)</em>) – Eigenvalues of operator T2.</p></li>
<li><p><strong>Rt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_t</em><em>)</em>) – Right eigenvectors of operator T2.</p></li>
<li><p><strong>Lt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_t</em><em>)</em>) – Left eigenvectors of operator T2.</p></li>
<li><p><strong>eta</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Weighting between spectral and Grassmann terms, default 0.5.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Exponent defining the OT ground cost. The returned cost is <span class="math notranslate nohighlight">\(d^p\)</span> with
<span class="math notranslate nohighlight">\(d^2 \approx C_2\)</span>. Default is 2.</p></li>
<li><p><strong>q</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Exponent applied to the eigenvalue distance in the spectral term.
Default is 1.</p></li>
<li><p><strong>grassmann_metric</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)"><em>str</em></a><em>, </em><em>optional</em>) – Metric type: “geodesic”, “chordal”, “procrustes”, or “martin”.</p></li>
<li><p><strong>eigen_scaling</strong> (<em>None</em><em> or </em><em>array-like</em><em> of </em><em>length 2</em><em>, </em><em>optional</em>) – Scaling <code class="docutils literal notranslate"><span class="pre">(real_scale,</span> <span class="pre">imag_scale)</span></code> applied to eigenvalues before computing
<span class="math notranslate nohighlight">\(C_\lambda\)</span>. If provided, eigenvalues are transformed as
<span class="math notranslate nohighlight">\(\lambda \mapsto \alpha\operatorname{Re}(\lambda) + i\,\beta\operatorname{Im}(\lambda)\)</span>.
If None, defaults to <code class="docutils literal notranslate"><span class="pre">(1.0,</span> <span class="pre">1.0)</span></code>. Accepts tuple/list or array/tensor with
two entries.</p></li>
<li><p><strong>nx</strong> (<em>module</em><em>, </em><em>optional</em>) – Backend (NumPy-compatible). If None, inferred from inputs.</p></li>
<li><p><strong>eps</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Minimum value used for numerical stability in Grassmann distances and
Martin metric. Default is 1e-12.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>C</strong> – SGOT cost matrix <span class="math notranslate nohighlight">\(C = d^p\)</span>.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>array-like, shape (n_s, n_t)</p>
</dd>
</dl>
<p class="rubric">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id1" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>83<span class="fn-bracket">]</span></span>
<p>Germain, T., Flamary, R., Kostic, V. R., & Lounici, K. (2025).
<a class="reference external" href="https://arxiv.org/abs/2509.24920">A Spectral-Grassmann Wasserstein Metric for Operator Representations
of Dynamical Systems</a>.</p>
</aside>
</aside>
</dd></dl>
<div class="sphx-glr-clear"></div><dl class="py function">
<dt class="sig sig-object py" id="ot.sgot.sgot_metric">
<span class="sig-prename descclassname"><span class="pre">ot.sgot.</span></span><span class="sig-name descname"><span class="pre">sgot_metric</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Ds</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Rs</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Ls</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Dt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Rt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Lt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eta</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">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">q</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">r</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">grassmann_metric</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'chordal'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eigen_scaling</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">Ws</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">Wt</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">nx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eps</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/sgot.html#sgot_metric"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sgot.sgot_metric" title="Link to this definition"></a></dt>
<dd><p>Compute the SGOT metric between two spectral decompositions.</p>
<p>This function computes a discrete optimal transport problem between two measures
over spectral atoms <span class="math notranslate nohighlight">\(z_i=(\lambda_i, V_i)\)</span> and <span class="math notranslate nohighlight">\(z'_j=(\lambda'_j, V'_j)\)</span>.
Using the ground cost matrix <span class="math notranslate nohighlight">\(C = d^p\)</span> returned by <a class="reference internal" href="#id4" title="ot.sgot.sgot_cost_matrix"><code class="xref py py-func docutils literal notranslate"><span class="pre">sgot_cost_matrix()</span></code></a>,
we solve:</p>
<div class="math notranslate nohighlight">
\[P^\star \in \arg\min_{P\in\Pi(W_s, W_t)} \langle C, P\rangle,\]</div>
<p>where <span class="math notranslate nohighlight">\(C(i,j) = d(i,j)^p\)</span> and <span class="math notranslate nohighlight">\(d(i,j)\)</span> is the SGOT ground distance
combining spectral and Grassmann terms with exponent <span class="math notranslate nohighlight">\(q\)</span>:</p>
<div class="math notranslate nohighlight">
\[d(i,j)^2
\;=\; \eta\,\big|\lambda_i - \lambda'_j\big|^{q}
\;+\; (1-\eta)\,d_G(i,j)^{q},\]</div>
<p>and <span class="math notranslate nohighlight">\(d_G(i,j)\)</span> is the Grassmann distance associated with the chosen
<code class="docutils literal notranslate"><span class="pre">grassmann_metric</span></code>.</p>
<p>From the optimal plan <span class="math notranslate nohighlight">\(P^\star\)</span>, we first form the <span class="math notranslate nohighlight">\(p\)</span>-Wasserstein
objective:</p>
<div class="math notranslate nohighlight">
\[\mathrm{obj}
\;=\;
\left(\sum_{i,j} C(i,j)\,P^\star_{ij}\right)^{1/p},\]</div>
<p>and then apply an outer root <span class="math notranslate nohighlight">\(r\)</span>:</p>
<div class="math notranslate nohighlight">
\[\mathrm{SGOT}
\;=\;
\mathrm{obj}^{1/r}.\]</div>
<p>In summary:</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(q\)</span> controls how strongly spectral and Grassmann distances are curved
(via <span class="math notranslate nohighlight">\(|\lambda_i - \lambda'_j|^{q}\)</span> and <span class="math notranslate nohighlight">\(d_G(i,j)^{q}\)</span>),</p></li>
<li><p><span class="math notranslate nohighlight">\(p\)</span> is the exponent used in the OT ground cost and the inner
Wasserstein root,</p></li>
<li><p><span class="math notranslate nohighlight">\(r\)</span> is an additional outer root applied to the Wasserstein objective.</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>Ds</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_s</em><em>,</em><em>)</em>) – Eigenvalues of operator T1.</p></li>
<li><p><strong>Rs</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_s</em><em>)</em>) – Right eigenvectors of operator T1.</p></li>
<li><p><strong>Ls</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_s</em><em>)</em>) – Left eigenvectors of operator T1.</p></li>
<li><p><strong>Dt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_t</em><em>,</em><em>)</em>) – Eigenvalues of operator T2.</p></li>
<li><p><strong>Rt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_t</em><em>)</em>) – Right eigenvectors of operator T2.</p></li>
<li><p><strong>Lt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_t</em><em>)</em>) – Left eigenvectors of operator T2.</p></li>
<li><p><strong>eta</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Weighting between spectral and Grassmann terms, default 0.5.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Exponent defining the OT ground cost, default 2.</p></li>
<li><p><strong>q</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Exponent applied to the eigenvalue and Grassmann distances, default 1.</p></li>
<li><p><strong>r</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em> or </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Outer root applied to the Wasserstein objective, default 2.</p></li>
<li><p><strong>grassmann_metric</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)"><em>str</em></a><em>, </em><em>optional</em>) – Metric type: “geodesic”, “chordal”, “procrustes”, or “martin”.</p></li>
<li><p><strong>eigen_scaling</strong> (<em>None</em><em> or </em><em>array-like</em><em> of </em><em>length 2</em><em>, </em><em>optional</em>) – Scaling <code class="docutils literal notranslate"><span class="pre">(real_scale,</span> <span class="pre">imag_scale)</span></code> applied to eigenvalues before
computing the spectral cost. If None, defaults to <code class="docutils literal notranslate"><span class="pre">(1.0,</span> <span class="pre">1.0)</span></code>.</p></li>
<li><p><strong>Ws</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_s</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Source weights. If None, uniform weights are used.</p></li>
<li><p><strong>Wt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_t</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Target weights. If None, uniform weights are used.</p></li>
<li><p><strong>nx</strong> (<em>module</em><em>, </em><em>optional</em>) – Backend (NumPy-compatible). If None, inferred from inputs.</p></li>
<li><p><strong>eps</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Numerical stability constant, default 1e-12.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>dist</strong> – SGOT distance between the two spectral decompositions.</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/library/functions.html#float" title="(in Python v3.14)">float</a></p>
</dd>
</dl>
<p class="rubric">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id2" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>83<span class="fn-bracket">]</span></span>
<p>Germain, T., Flamary, R., Kostic, V. R., & Lounici, K. (2025).
<a class="reference external" href="https://arxiv.org/abs/2509.24920">A Spectral-Grassmann Wasserstein Metric for Operator Representations
of Dynamical Systems</a>.</p>
</aside>
</aside>
</dd></dl>
<section id="examples-using-ot-sgot-sgot-metric">
<h3>Examples using <code class="docutils literal notranslate"><span class="pre">ot.sgot.sgot_metric</span></code><a class="headerlink" href="#examples-using-ot-sgot-sgot-metric" title="Link to this heading"></a></h3>
<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="This example presents a synthetic example of Spectral Grassmannian-Wasserstein Optimal Transport (SGOT) on linear dynamical systems."><img alt="" src="../_images/sphx_glr_plot_sgot_thumb.png" />
<p><a class="reference internal" href="../auto_examples/others/plot_sgot.html"><span class="doc">Spectral-Grassmann OT on dynamical systems operators</span></a></p>
<div class="sphx-glr-thumbnail-title">Spectral-Grassmann OT on dynamical systems operators</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.sgot.</span></span><span class="sig-name descname"><span class="pre">eigenvalue_cost_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Ds</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Dt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">q</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eigen_scaling</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">nx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/sgot.html#eigenvalue_cost_matrix"><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 pairwise eigenvalue distances for source and target domains.</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>Ds</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_s</em><em>,</em><em>)</em>) – Source eigenvalues.</p></li>
<li><p><strong>Dt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_t</em><em>,</em><em>)</em>) – Target eigenvalues.</p></li>
<li><p><strong>q</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em> or </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Exponent applied to the eigenvalue distance, default 1.</p></li>
<li><p><strong>eigen_scaling</strong> (<em>None</em><em> or </em><em>array-like</em><em> of </em><em>length 2</em><em>, </em><em>optional</em>) – Scaling (real_scale, imag_scale) applied to eigenvalues before computing
distances. If None, defaults to (1.0, 1.0). Accepts tuple/list or
array/tensor with two entries.</p></li>
<li><p><strong>nx</strong> (<em>module</em><em>, </em><em>optional</em>) – Backend (NumPy-compatible). If None, inferred from inputs.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>C</strong> – Eigenvalue cost matrix.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>np.ndarray, shape (n_s, n_t)</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="id4">
<span class="sig-prename descclassname"><span class="pre">ot.sgot.</span></span><span class="sig-name descname"><span class="pre">sgot_cost_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Ds</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Rs</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Ls</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Dt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Rt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Lt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eta</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">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">q</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">grassmann_metric</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'chordal'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eigen_scaling</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">nx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eps</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/sgot.html#sgot_cost_matrix"><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>Compute the SGOT cost matrix between two spectral decompositions.</p>
<p>This returns the discrete ground cost matrix used in the SGOT optimal transport
objective. Each spectral atom is <span class="math notranslate nohighlight">\(z_i=(\lambda_i, V_i)\)</span> where
<span class="math notranslate nohighlight">\(\lambda_i \in \mathbb{C}\)</span> is an eigenvalue and <span class="math notranslate nohighlight">\(V_i\)</span> is the
associated (bi-orthogonal) eigenspace point.</p>
<div class="math notranslate nohighlight">
\[C_2(i,j) \;=\; \eta\,C_\lambda(i,j) \;+\; (1-\eta)\,C_G(i,j),\]</div>
<p>with spectral term</p>
<div class="math notranslate nohighlight">
\[C_\lambda(i,j) \;=\; \big|\lambda_i - \lambda'_j\big|^{q},\]</div>
<p>and Grassmann term computed from a similarity score <span class="math notranslate nohighlight">\(\delta_{ij}\in[0,1]\)</span>
built from left/right eigenvectors</p>
<div class="math notranslate nohighlight">
\[\delta_{ij} \;=\; \left|\langle r_i, r'_j\rangle\,\langle \ell_i, \ell'_j\rangle\right|.\]</div>
<p>Depending on <code class="docutils literal notranslate"><span class="pre">grassmann_metric</span></code>, the Grassmann contribution is:</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">"chordal"</span></code>: <span class="math notranslate nohighlight">\(C_G(i,j) = 1 - \delta_{ij}^2\)</span></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"geodesic"</span></code>: <span class="math notranslate nohighlight">\(C_G(i,j) = \arccos(\delta_{ij})^2\)</span></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"procrustes"</span></code>: <span class="math notranslate nohighlight">\(C_G(i,j) = 2(1-\delta_{ij})\)</span></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"martin"</span></code>: <span class="math notranslate nohighlight">\(C_G(i,j) = -\log\!\left(\max(\delta_{ij}^2,\varepsilon)\right)\)</span></p></li>
</ul>
<p>Finally, we return a matrix suited for a <span class="math notranslate nohighlight">\(p\)</span>-Wasserstein objective by
treating <span class="math notranslate nohighlight">\(C_2 \approx d^2\)</span> and outputting</p>
<div class="math notranslate nohighlight">
\[C(i,j) \;=\; \big(\operatorname{Re}(C_2(i,j))\big)^{p/2}.\]</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>Ds</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_s</em><em>,</em><em>)</em>) – Eigenvalues of operator T1.</p></li>
<li><p><strong>Rs</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_s</em><em>)</em>) – Right eigenvectors of operator T1.</p></li>
<li><p><strong>Ls</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_s</em><em>)</em>) – Left eigenvectors of operator T1.</p></li>
<li><p><strong>Dt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_t</em><em>,</em><em>)</em>) – Eigenvalues of operator T2.</p></li>
<li><p><strong>Rt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_t</em><em>)</em>) – Right eigenvectors of operator T2.</p></li>
<li><p><strong>Lt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_t</em><em>)</em>) – Left eigenvectors of operator T2.</p></li>
<li><p><strong>eta</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Weighting between spectral and Grassmann terms, default 0.5.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Exponent defining the OT ground cost. The returned cost is <span class="math notranslate nohighlight">\(d^p\)</span> with
<span class="math notranslate nohighlight">\(d^2 \approx C_2\)</span>. Default is 2.</p></li>
<li><p><strong>q</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Exponent applied to the eigenvalue distance in the spectral term.
Default is 1.</p></li>
<li><p><strong>grassmann_metric</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)"><em>str</em></a><em>, </em><em>optional</em>) – Metric type: “geodesic”, “chordal”, “procrustes”, or “martin”.</p></li>
<li><p><strong>eigen_scaling</strong> (<em>None</em><em> or </em><em>array-like</em><em> of </em><em>length 2</em><em>, </em><em>optional</em>) – Scaling <code class="docutils literal notranslate"><span class="pre">(real_scale,</span> <span class="pre">imag_scale)</span></code> applied to eigenvalues before computing
<span class="math notranslate nohighlight">\(C_\lambda\)</span>. If provided, eigenvalues are transformed as
<span class="math notranslate nohighlight">\(\lambda \mapsto \alpha\operatorname{Re}(\lambda) + i\,\beta\operatorname{Im}(\lambda)\)</span>.
If None, defaults to <code class="docutils literal notranslate"><span class="pre">(1.0,</span> <span class="pre">1.0)</span></code>. Accepts tuple/list or array/tensor with
two entries.</p></li>
<li><p><strong>nx</strong> (<em>module</em><em>, </em><em>optional</em>) – Backend (NumPy-compatible). If None, inferred from inputs.</p></li>
<li><p><strong>eps</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Minimum value used for numerical stability in Grassmann distances and
Martin metric. Default is 1e-12.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>C</strong> – SGOT cost matrix <span class="math notranslate nohighlight">\(C = d^p\)</span>.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>array-like, shape (n_s, n_t)</p>
</dd>
</dl>
<p class="rubric">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id5" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>83<span class="fn-bracket">]</span></span>
<p>Germain, T., Flamary, R., Kostic, V. R., & Lounici, K. (2025).
<a class="reference external" href="https://arxiv.org/abs/2509.24920">A Spectral-Grassmann Wasserstein Metric for Operator Representations
of Dynamical Systems</a>.</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.sgot.</span></span><span class="sig-name descname"><span class="pre">sgot_metric</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Ds</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Rs</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Ls</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Dt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Rt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Lt</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eta</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">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">q</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">r</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">grassmann_metric</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'chordal'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eigen_scaling</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">Ws</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">Wt</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">nx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eps</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/sgot.html#sgot_metric"><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 the SGOT metric between two spectral decompositions.</p>
<p>This function computes a discrete optimal transport problem between two measures
over spectral atoms <span class="math notranslate nohighlight">\(z_i=(\lambda_i, V_i)\)</span> and <span class="math notranslate nohighlight">\(z'_j=(\lambda'_j, V'_j)\)</span>.
Using the ground cost matrix <span class="math notranslate nohighlight">\(C = d^p\)</span> returned by <a class="reference internal" href="#id4" title="ot.sgot.sgot_cost_matrix"><code class="xref py py-func docutils literal notranslate"><span class="pre">sgot_cost_matrix()</span></code></a>,
we solve:</p>
<div class="math notranslate nohighlight">
\[P^\star \in \arg\min_{P\in\Pi(W_s, W_t)} \langle C, P\rangle,\]</div>
<p>where <span class="math notranslate nohighlight">\(C(i,j) = d(i,j)^p\)</span> and <span class="math notranslate nohighlight">\(d(i,j)\)</span> is the SGOT ground distance
combining spectral and Grassmann terms with exponent <span class="math notranslate nohighlight">\(q\)</span>:</p>
<div class="math notranslate nohighlight">
\[d(i,j)^2
\;=\; \eta\,\big|\lambda_i - \lambda'_j\big|^{q}
\;+\; (1-\eta)\,d_G(i,j)^{q},\]</div>
<p>and <span class="math notranslate nohighlight">\(d_G(i,j)\)</span> is the Grassmann distance associated with the chosen
<code class="docutils literal notranslate"><span class="pre">grassmann_metric</span></code>.</p>
<p>From the optimal plan <span class="math notranslate nohighlight">\(P^\star\)</span>, we first form the <span class="math notranslate nohighlight">\(p\)</span>-Wasserstein
objective:</p>
<div class="math notranslate nohighlight">
\[\mathrm{obj}
\;=\;
\left(\sum_{i,j} C(i,j)\,P^\star_{ij}\right)^{1/p},\]</div>
<p>and then apply an outer root <span class="math notranslate nohighlight">\(r\)</span>:</p>
<div class="math notranslate nohighlight">
\[\mathrm{SGOT}
\;=\;
\mathrm{obj}^{1/r}.\]</div>
<p>In summary:</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(q\)</span> controls how strongly spectral and Grassmann distances are curved
(via <span class="math notranslate nohighlight">\(|\lambda_i - \lambda'_j|^{q}\)</span> and <span class="math notranslate nohighlight">\(d_G(i,j)^{q}\)</span>),</p></li>
<li><p><span class="math notranslate nohighlight">\(p\)</span> is the exponent used in the OT ground cost and the inner
Wasserstein root,</p></li>
<li><p><span class="math notranslate nohighlight">\(r\)</span> is an additional outer root applied to the Wasserstein objective.</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>Ds</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_s</em><em>,</em><em>)</em>) – Eigenvalues of operator T1.</p></li>
<li><p><strong>Rs</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_s</em><em>)</em>) – Right eigenvectors of operator T1.</p></li>
<li><p><strong>Ls</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_s</em><em>)</em>) – Left eigenvectors of operator T1.</p></li>
<li><p><strong>Dt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_t</em><em>,</em><em>)</em>) – Eigenvalues of operator T2.</p></li>
<li><p><strong>Rt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_t</em><em>)</em>) – Right eigenvectors of operator T2.</p></li>
<li><p><strong>Lt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>L</em><em>, </em><em>n_t</em><em>)</em>) – Left eigenvectors of operator T2.</p></li>
<li><p><strong>eta</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Weighting between spectral and Grassmann terms, default 0.5.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Exponent defining the OT ground cost, default 2.</p></li>
<li><p><strong>q</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Exponent applied to the eigenvalue and Grassmann distances, default 1.</p></li>
<li><p><strong>r</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em> or </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Outer root applied to the Wasserstein objective, default 2.</p></li>
<li><p><strong>grassmann_metric</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)"><em>str</em></a><em>, </em><em>optional</em>) – Metric type: “geodesic”, “chordal”, “procrustes”, or “martin”.</p></li>
<li><p><strong>eigen_scaling</strong> (<em>None</em><em> or </em><em>array-like</em><em> of </em><em>length 2</em><em>, </em><em>optional</em>) – Scaling <code class="docutils literal notranslate"><span class="pre">(real_scale,</span> <span class="pre">imag_scale)</span></code> applied to eigenvalues before
computing the spectral cost. If None, defaults to <code class="docutils literal notranslate"><span class="pre">(1.0,</span> <span class="pre">1.0)</span></code>.</p></li>
<li><p><strong>Ws</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_s</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Source weights. If None, uniform weights are used.</p></li>
<li><p><strong>Wt</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>n_t</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Target weights. If None, uniform weights are used.</p></li>
<li><p><strong>nx</strong> (<em>module</em><em>, </em><em>optional</em>) – Backend (NumPy-compatible). If None, inferred from inputs.</p></li>
<li><p><strong>eps</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Numerical stability constant, default 1e-12.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>dist</strong> – SGOT distance between the two spectral decompositions.</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/library/functions.html#float" title="(in Python v3.14)">float</a></p>
</dd>
</dl>
<p class="rubric">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id8" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>83<span class="fn-bracket">]</span></span>
<p>Germain, T., Flamary, R., Kostic, V. R., & Lounici, K. (2025).
<a class="reference external" href="https://arxiv.org/abs/2509.24920">A Spectral-Grassmann Wasserstein Metric for Operator Representations
of Dynamical Systems</a>.</p>
</aside>
</aside>
</dd></dl>
</section>
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