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<!DOCTYPE html>

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<title>ot.batch — POT Python Optimal Transport 0.9.7 documentation</title>

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POT Python Optimal Transport

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<li class="toctree-l3"><a class="reference internal" href="#ot.batch.bregman_log_projection_batch"><code class="docutils literal notranslate">bregman_log_projection_batch()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#ot.batch.bregman_projection_batch"><code class="docutils literal notranslate">bregman_projection_batch()</code></a></li>

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<h1>ot.batch<a class="headerlink" href="#module-ot.batch" title="Link to this heading"></a></h1>

Batch operations for optimal transport.

ot.batch.bregman_log_projection_batch(K, a=None, b=None, nx=None, max_iter=10000, tol=1e-05, grad='detach')<a class="reference internal" href="../_modules/ot/batch/_utils.html#bregman_log_projection_batch">[source]</a><a class="headerlink" href="#ot.batch.bregman_log_projection_batch" title="Link to this definition"></a></dt>

<dd>Apply Bregman projection to a batch of affinity matrices \(\exp(\mathbf{K})\).

The function solves the following optimization problem:

\[\begin{split}\begin{aligned}

\mathbf{T} = \mathop{\arg \min}_\mathbf{T} \quad & \text{KL}(\mathbf{T} \| \exp(\mathbf{K})) \\

\text{s.t.} \quad & \mathbf{T} \mathbf{1} = \mathbf{a} \\

& \mathbf{T}^T \mathbf{1} = \mathbf{b} \\

& \mathbf{T} \geq 0

\end{aligned}\end{split}\]</div>

This is equivalent to:

\[\begin{split}\begin{aligned}

\mathbf{T} = \mathop{\arg \max}_\mathbf{T} \quad & \langle \mathbf{T}, \mathbf{K} \rangle_F \\

\text{s.t.} \quad & \mathbf{T} \mathbf{1} = \mathbf{a} \\

& \mathbf{T}^T \mathbf{1} = \mathbf{b} \\

& \mathbf{T} \geq 0

\end{aligned}\end{split}\]</div>

The optimal solution has the form \(\mathbf{T} = \text{diag}(\exp(\mathbf{u})) \exp(\mathbf{K}) \text{diag}(\exp(\mathbf{v}))\),

where the dual variables \(\mathbf{u}\) and \(\mathbf{v}\) are found iteratively using:

\[ \begin{align}\begin{aligned}\mathbf{u}^{(k+1)} = \log(\mathbf{a}) - \text{LSE}(\mathbf{K} + \mathbf{v}^{(k)})\\\mathbf{v}^{(k+1)} = \log(\mathbf{b}) - \text{LSE}(\mathbf{K}^T + \mathbf{u}^{(k)})\end{aligned}\end{align} \]</div>

where LSE denotes the log-sum-exp operation. The iterations are performed using

the log-sum-exp trick to avoid numerical issues.

<dt class="field-odd">Parameters:</dt>

<li>K (array-like, shape (B, n, m)) – Affinity matrix for each problem in the batch.</li>

<li>a (array-like, shape (B, n), optional) – Source distribution for each problem. If None, uniform distribution is used.</li>

<li>b (array-like, shape (B, m), optional) – Target distribution for each problem. If None, uniform distribution is used.</li>

<li>nx (backend object, optional) – Numerical backend to use for computations. If None, the default backend is used.</li>

<li>max_iter (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)">int</a>, optional) – Maximum number of iterations.</li>

<li>tol (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>, optional) – Tolerance for convergence. The solver stops when the maximum change in

the dual variables is below this value.</li>

<li>grad (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)">str</a>, optional) – Gradient computation mode: ‘detach’, ‘autodiff’, or ‘last_step’.</li>

</ul>

</dd>

<dt class="field-even">Returns:</dt>

<dd class="field-even">Dictionary containing:

- ‘T’ : array-like, shape (B, n, m)

<div>Optimal transport plan for each problem.

</div></blockquote>

<dt>’log_T’array-like, shape (B, n, m)</dt><dd>Logarithm of the optimal transport plan for each problem.

</dd>

</dl>

</li>

<dt>’potentials’tuple of array-like, shapes ((B, n), (B, m))</dt><dd>Log-scaling factors (u, v).

</dd>

</dl>

</li>

<dt>’n_iters’int</dt><dd>Number of iterations performed.

</dd>

</dl>

</li>

</ul>

</dd>

<dt class="field-odd">Return type:</dt>

</dd>

</dl>

Examples

<div class="doctest highlight-default notranslate"><div class="highlight"><pre>>>> import numpy as np

>>> from ot.batch import bregman_log_projection_batch

>>> # Create batch of affinity matrices

>>> K = np.random.randn(5, 10, 15) # 5 problems, 10x15 cost matrices

>>> result = bregman_log_projection_batch(K)

>>> T = result['T'] # Shape (5, 10, 15)

</pre></div>

</div>

See also

<dt><a class="reference internal" href="#ot.batch.bregman_projection_batch" title="ot.batch.bregman_projection_batch"><code class="xref py py-obj docutils literal notranslate">ot.batch.bregman_projection_batch</code></a></dt><dd>standard Bregman projection.

</dd>

</dl>

</div>

</dd></dl>

ot.batch.bregman_projection_batch(K, a=None, b=None, nx=None, max_iter=10000, tol=1e-05, grad='detach')<a class="reference internal" href="../_modules/ot/batch/_utils.html#bregman_projection_batch">[source]</a><a class="headerlink" href="#ot.batch.bregman_projection_batch" title="Link to this definition"></a></dt>

<dd>Apply Bregman projection to a batch of affinity matrices \(\mathbf{K}\).

The function solves the following optimization problem:

\[\begin{split}\begin{aligned}

\mathbf{T} = \mathop{\arg \min}_\mathbf{T} \quad & \text{KL}(\mathbf{T} \| \mathbf{K}) \\

\text{s.t.} \quad & \mathbf{T} \mathbf{1} = \mathbf{a} \\

& \mathbf{T}^T \mathbf{1} = \mathbf{b} \\

& \mathbf{T} \geq 0

\end{aligned}\end{split}\]</div>

This is equivalent to:

\[\begin{split}\begin{aligned}

\mathbf{T} = \mathop{\arg \max}_\mathbf{T} \quad & \langle \mathbf{T}, \log(\mathbf{K}) \rangle_F \\

\text{s.t.} \quad & \mathbf{T} \mathbf{1} = \mathbf{a} \\

& \mathbf{T}^T \mathbf{1} = \mathbf{b} \\

& \mathbf{T} \geq 0

\end{aligned}\end{split}\]</div>

The optimal solution has the form \(\mathbf{T} = \text{diag}(\mathbf{f}) \mathbf{K} \text{diag}(\mathbf{g})\),

where the dual variables \(\mathbf{u}\) and \(\mathbf{v}\) are found iteratively using:

\[ \begin{align}\begin{aligned}\mathbf{f}^{(k+1)} = \frac{\mathbf{a}}{\sum \mathbf{K} \mathbf{g}^{(k)}}\\\mathbf{g}^{(k+1)} = \frac{\mathbf{b}}{\sum \mathbf{K}^T \mathbf{f}^{(k)}}\end{aligned}\end{align} \]</div>

<dt class="field-odd">Parameters:</dt>

<li>nx (backend object, optional) – Numerical backend to use for computations. If None, the default backend is used.</li>

the dual variables is below this value.</li>

</ul>

</dd>

<dt class="field-even">Returns:</dt>

<dd class="field-even">Dictionary containing:

- ‘T’ : array-like, shape (B, n, m)

<div>Optimal transport plan for each problem.

</div></blockquote>

<dt>’potentials’tuple of array-like, shapes ((B, n), (B, m))</dt><dd>Scaling factors (f, g).

</dd>

</dl>

</li>

<dt>’n_iters’int</dt><dd>Number of iterations performed.

</dd>

</dl>

</li>

</ul>

</dd>

<dt class="field-odd">Return type:</dt>

</dd>

</dl>

Examples

>>> # Create batch of affinity matrices

>>> result = bregman_projection_batch(K)

</pre></div>

</div>

See also

<dt><a class="reference internal" href="#ot.batch.bregman_log_projection_batch" title="ot.batch.bregman_log_projection_batch"><code class="xref py py-obj docutils literal notranslate">ot.batch.bregman_log_projection_batch</code></a></dt><dd>Bregman projection in the log-domain.

</dd>

</dl>

</div>

</dd></dl>

ot.batch.dist_batch(X1, X2=None, metric='sqeuclidean', p=2, nx=None)<a class="reference internal" href="../_modules/ot/batch/_linear.html#dist_batch">[source]</a><a class="headerlink" href="#ot.batch.dist_batch" title="Link to this definition"></a></dt>

<dd>Batched version of ot.dist, use it to compute many distance matrices in parallel.

<dt class="field-odd">Parameters:</dt>

<li>X1 (array-like, shape (<a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b">b</a>,n1,d)) – <cite>b</cite> matrices with <cite>n1</cite> samples of size <cite>d</cite></li>

<li>X2 (array-like, shape (<a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b">b</a>,n2,d), optional) – <cite>b</cite> matrices with <cite>n2</cite> samples of size <cite>d</cite> (if None then \(\mathbf{X_2} = \mathbf{X_1}\))</li>

<li>metric (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)">str</a>, optional) – ‘sqeuclidean’, ‘euclidean’, ‘minkowski’ or ‘kl’</li>

<li>p (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>, optional) – p-norm for the Minkowski metrics. Default value is 2.</li>

<li>nx (<a class="reference internal" href="ot.backend.html#id0" title="ot.backend.Backend">Backend</a>, optional) – Backend to perform computations on. If omitted, the backend defaults to that of <cite>x1</cite>.</li>

</ul>

</dd>

<dt class="field-even">Returns:</dt>

<dd class="field-even">M – distance matrix computed with given metric

</dd>

<dt class="field-odd">Return type:</dt>

<dd class="field-odd">array-like, shape (<cite>b</cite>, <cite>n1</cite>, <cite>n2</cite>)

</dd>

</dl>

Examples

>>> X1 = np.random.randn(5, 10, 3)

>>> X2 = np.random.randn(5, 15, 3)

>>> M = dist_batch(X1, X2, metric="euclidean")

>>> M.shape

(5, 10, 15)

</pre></div>

</div>

See also

<dt><a class="reference internal" href="../all.html#ot.dist" title="ot.dist"><code class="xref py py-obj docutils literal notranslate">ot.dist</code></a></dt><dd>equivalent non-batched function.

</dd>

</dl>

</div>

</dd></dl>

ot.batch.entropy_batch(T, nx=None, eps=1e-16)<a class="reference internal" href="../_modules/ot/batch/_utils.html#entropy_batch">[source]</a><a class="headerlink" href="#ot.batch.entropy_batch" title="Link to this definition"></a></dt>

<dd>Computes the entropy of a batch of transport plans T.

\[H(T)_b = - \sum_{i,j} T_{b,i,j} \log(T_{b,i,j})\]</div>

<dt class="field-odd">Parameters:</dt>

<li>T (array-like, shape (<a class="reference internal" href="ot.utils.html#id35" title="ot.utils.BaryResult.b">b</a>,n1,n2)) – <cite>b</cite> transport of size <cite>n1</cite> x <cite>n2</cite>.</li>

<li>eps (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>, optional) – Small constant to avoid numerical issues with log(0). Default is 1e-16.</li>

</ul>

</dd>

</dl>

</dd></dl>

ot.batch.loss_linear_batch(M, T, nx=None)<a class="reference internal" href="../_modules/ot/batch/_linear.html#loss_linear_batch">[source]</a><a class="headerlink" href="#ot.batch.loss_linear_batch" title="Link to this definition"></a></dt>

<dd>Computes the linear optimal transport loss given a batch cost matrices and transport plans.

\[L(T, M)_b = \langle T_b, M_b \rangle_F\]</div>

<dt class="field-odd">Parameters:</dt>

<li>M (array-like, shape (B, ns, nt)) – Cost matrix</li>

<li>T (array-like, shape (B, ns, nt)) – Transport plan</li>

</ul>

</dd>

<dt class="field-even">Returns:</dt>

<dd class="field-even">loss – Loss value for each batch element

</dd>

<dt class="field-odd">Return type:</dt>

<dd class="field-odd">array-like, shape (B,)

</dd>

</dl>

See also

<dt><a class="reference internal" href="#ot.batch.dist_batch" title="ot.batch.dist_batch"><code class="xref py py-obj docutils literal notranslate">ot.batch.dist_batch</code></a></dt><dd>batched cost matrix computation for computing M.

</dd>

<dt><a class="reference internal" href="#ot.batch.solve_batch" title="ot.batch.solve_batch"><code class="xref py py-obj docutils literal notranslate">ot.batch.solve_batch</code></a></dt><dd>solver for computing the optimal T.

</dd>

</dl>

</div>

</dd></dl>

ot.batch.loss_linear_samples_batch(X, Y, T, metric='sqeuclidean')<a class="reference internal" href="../_modules/ot/batch/_linear.html#loss_linear_samples_batch">[source]</a><a class="headerlink" href="#ot.batch.loss_linear_samples_batch" title="Link to this definition"></a></dt>

<dd>Computes the linear optimal transport loss given samples and transport plan. This is the equivalent of

calling <cite>dist_batch</cite> and then <cite>loss_linear_batch</cite>.

<dt class="field-odd">Parameters:</dt>

<li>X (array-like, shape (B, ns, d)) – Samples from source distribution</li>

<li>Y (array-like, shape (B, nt, d)) – Samples from target distribution</li>

<li>T (array-like, shape (B, ns, nt)) – Transport plan</li>

</ul>

</dd>

<dt class="field-even">Returns:</dt>

<dd class="field-even">loss – Loss value for each batch element

</dd>

<dt class="field-odd">Return type:</dt>

<dd class="field-odd">array-like, shape (B,)

</dd>

</dl>

See also

</dd>

</dl>

</div>

</dd></dl>

ot.batch.loss_quadratic_batch(a, b, Ca, Cb, T, M=None, alpha=None, unbalanced=None, unbalanced_type='kl', loss='sqeuclidean', symmetric=True, nx=None, logits=None, recompute_const=False, log=False)<a class="reference internal" href="../_modules/ot/batch/_quadratic.html#loss_quadratic_batch">[source]</a><a class="headerlink" href="#ot.batch.loss_quadratic_batch" title="Link to this definition"></a></dt>

<dd>Computes the gromov-wasserstein for samples Ca, Cb and transport plan. Batched version.

<dt class="field-odd">Parameters:</dt>

<li>a (array-like, shape (B, n)) – Source distributions.</li>

<li>b (array-like, shape (B, m)) – Target distributions.</li>

<li>Ca (array-like, shape (B, n, n) or (B, n, n, d)) – Source cost matrices.</li>

<li>Cb (array-like, shape (B, m, m) or (B, n, n, d)) – Target cost matrices.</li>

<li>T (array-like, shape (B, n, m)) – Transport plan.</li>

<li>M (array-like, shape (B, n, m)) – Cost matrix between features across domains (default is None).</li>

<li>alpha (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>, array-like or <a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#list" title="(in Python v3.14)">list</a> (B,) optional) – Weight the quadratic term (alpha*Gromov) and the linear term

((1-alpha)*Wass) in the Fused Gromov-Wasserstein problem. Not used for

Gromov problem (when M is not provided). By default <code class="docutils literal notranslate">alpha=None</code>

corresponds to <code class="docutils literal notranslate">alpha=1</code> for Gromov problem (<code class="docutils literal notranslate">M==None</code>) and

<code class="docutils literal notranslate">alpha=0.5</code> for Fused Gromov-Wasserstein problem (<code class="docutils literal notranslate">M!=None</code>).

If alpha is a scalar, it is used for all problems in the batch.</li>

<li>unbalanced (float array-like or <a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#list" title="(in Python v3.14)">list</a>(B,) optional) – Unbalanced penalization weight \(\lambda_u\). If unbalanced is a scalar, it is used for all problems in the batch.</li>

<li>unbalanced_type (string, optional) – Type of unbalanced penalization function, either “kl” (Kullback-Leibler divergence) or “l2” (half-squared L2 divergence)</li>

<li>loss (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)">str</a>, optional) – Loss function to use. Supported values: ‘sqeuclidean’, ‘kl’.

Default is ‘sqeuclidean’.</li>

<li>symmetric (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)">bool</a>, optional) – Whether to use symmetric version. Default is True.</li>

<li>nx (module, optional) – Backend to use. Default is None.</li>

<li>logits (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)">bool</a>, optional) – For KL divergence, whether inputs are logits (unnormalized log probabilities).

If True, inputs are treated as logits. Default is None.</li>

<li>recompute_const (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)">bool</a>, optional) – Whether to recompute the constant term. Default is False. This should be set to True if T does not satisfy the marginal constraints.

Will be set to True if unbalanced is not None.</li>

<li>log (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)">bool</a>, optional) – If True, also returns a dictionary containing the different terms of

the loss.</li>

</ul>

</dd>

</dl>

Examples

>>> # Create batch of cost matrices

>>> Ca = np.random.rand(3, 5, 5) # 3 problems, 5x5 source matrices

>>> Cb = np.random.rand(3, 4, 4) # 3 problems, 4x4 target matrices

>>> a = np.ones((3, 5)) / 5 # Uniform source distributions

>>> b = np.ones((3, 4)) / 4 # Uniform target distributions

>>> # Use the uniform transport plan for testing

>>> T = np.ones((3, 5, 4)) / (5 * 4)

>>> loss = loss_quadratic_samples_batch(a, b, Ca, Cb, T, recompute_const=True)

>>> loss.shape

(3,)

</pre></div>

</div>

See also

<dt><a class="reference internal" href="#ot.batch.tensor_batch" title="ot.batch.tensor_batch"><code class="xref py py-obj docutils literal notranslate">ot.batch.tensor_batch</code></a></dt><dd>From computing the cost tensor L.

</dd>

<dt><a class="reference internal" href="#ot.batch.solve_gromov_batch" title="ot.batch.solve_gromov_batch"><code class="xref py py-obj docutils literal notranslate">ot.batch.solve_gromov_batch</code></a></dt><dd>For finding the optimal transport plan T.

</dd>

</dl>

</div>

</dd></dl>

ot.batch.loss_quadratic_samples_batch(*args, **kwargs)<a class="headerlink" href="#ot.batch.loss_quadratic_samples_batch" title="Link to this definition"></a></dt>

<dd>DEPRECATED: Use ot.batch.loss_quadratic_batch instead.

</dd></dl>

ot.batch.proximal_bregman_log_plan_batch(C, a=None, b=None, nx=None, reg=None, inner_reg=0.001, max_iter=10000, tol=1e-05, inner_iter=1, grad='detach')<a class="reference internal" href="../_modules/ot/batch/_utils.html#proximal_bregman_log_plan_batch">[source]</a><a class="headerlink" href="#ot.batch.proximal_bregman_log_plan_batch" title="Link to this definition"></a></dt>

<dd>Returns optimal transport plans for a batch of cost matrices \(\mathbf{C}\) using a Bregman divergence based proximal point method <a class="reference internal" href="#references-ot-bregman-proximal-point">[92]</a>.

This function solves the following optimization problem:

\[\begin{split}\begin{aligned}

\mathbf{T} = \mathop{\arg \min}_\mathbf{T} \quad & \langle \mathbf{T}, \mathbf{C} \rangle_F + \textit{reg} \cdot \sum_{i,j} \mathbf{T}_{i,j} \log \mathbf{T}_{i,j} \\

\text{s.t.} \quad & \mathbf{T} \mathbf{1} = \mathbf{a} \\

& \mathbf{T}^T \mathbf{1} = \mathbf{b} \\

& \mathbf{T} \geq 0

\end{aligned}\end{split}\]</div>

The optimal transport plans are computed iteratively with a proximal point method based on a Bregman divergence where each iteration involves solving a Bregman projection problem:

\[\mathbf{T}^{(k+1)} = \mathop{\arg \min}_\mathbf{T} \quad \langle \mathbf{C} - \textit{inner\_reg} \cdot \log \mathbf{T}^{(k)}, \mathbf{T} \rangle + (\textit{reg} + \textit{inner\_reg}) \cdot \sum_{i,j} \mathbf{T}_{i,j} \log \mathbf{T}_{i,j}\]</div>

Denoting \(\mathbf{K}^{(k)} = - (\mathbf{C} + \textit{inner\_reg} \cdot \log \mathbf{T}^{(k)})/(\textit{reg} + \textit{inner\_reg})\), the affinity matrix at iteration \(k\), the Bregman projection problem is solved in the log-domain with a finite number of inner iterations \(\text{inner\_iter}\), i.e., the dual variables \(\mathbf{u}\) and \(\mathbf{v}\) are updated as follows:

\[ \begin{align}\begin{aligned}\mathbf{u}^{(i+1)} = \log(\mathbf{a}) - \text{LSE}(\mathbf{K}^{(k)} + \mathbf{v}^{(i)})\\\mathbf{v}^{(i+1)} = \log(\mathbf{b}) - \text{LSE}((\mathbf{K}^{(k)})^T + \mathbf{u}^{(i)})\end{aligned}\end{align} \]</div>

where LSE denotes the log-sum-exp operation.

<dt class="field-odd">Parameters:</dt>

<li>C (array-like, shape (B, n, m)) – Cost matrix for each problem in the batch.</li>

<li>nx (backend object, optional) – Numerical backend to use for computations. If None, the default backend is used.</li>

<li>inner_iter (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)">int</a>, optional) – Number of inner iterations for updating the dual variables u and v. Default is 1.</li>

<li>inner_reg (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>, optional) – Regularization parameter for the Bregman divergence. Default is 1e-3.</li>

the dual variables is below this value.</li>

</ul>

</dd>

<dt class="field-even">Returns:</dt>

<dd class="field-even">Dictionary containing:

- ‘T’ : array-like, shape (B, n, m)

<div>Optimal transport plan for each problem.

</div></blockquote>

<dt>’log_T’array-like, shape (B, n, m)</dt><dd>Logarithm of the optimal transport plan for each problem.

</dd>

</dl>

</li>

<dt>’potentials’tuple of array-like, shapes ((B, n), (B, m))</dt><dd>Log-scaling factors (u, v).

</dd>

</dl>

</li>

<dt>’n_iters’int</dt><dd>Number of iterations performed.

</dd>

</dl>

</li>

</ul>

</dd>

<dt class="field-odd">Return type:</dt>

</dd>

</dl>

Example

>>> # Create batch of affinity matrices

>>> C = np.random.rand(5, 10, 15) # 5 problems, 10x15 cost matrices

>>> result = proximal_bregman_log_plan_batch(C)

</pre></div>

</div>

<h2>Reference<a class="headerlink" href="#reference" title="Link to this heading"></a></h2>

[92]

Xie, Y., Wang, X., Wang, R., & Zha, H. (2020, August).

</aside>

A fast proximal point method for computing exact wasserstein distance.

In Uncertainty in artificial intelligence (pp. 433-453). PMLR.

</section>

</dd></dl>

ot.batch.solve_batch(M, reg=None, a=None, b=None, max_iter=1000, tol=1e-05, method='auto', inner_iter=1, inner_reg=0.001, reg_type='entropy', grad='envelope')<a class="reference internal" href="../_modules/ot/batch/_linear.html#solve_batch">[source]</a><a class="headerlink" href="#ot.batch.solve_batch" title="Link to this definition"></a></dt>

<dd>Return solutions of a batch of discrete optimal transport problems in a <a class="reference internal" href="ot.utils.html#id56" title="ot.utils.OTResult"><code class="xref any py py-class docutils literal notranslate">OTResult</code></a> object.

The function solves in parallel a batch of optimal transport problems:

\[\begin{split}\begin{aligned}

\mathbf{T} = \mathop{\arg \min}_\mathbf{T} \quad & \langle \mathbf{T}, \mathbf{M} \rangle_F + \textit{reg} \cdot R(\mathbf{T}) \\

\text{s.t.} \quad & \mathbf{T} \mathbf{1} = \mathbf{a} \\

& \mathbf{T}^T \mathbf{1} = \mathbf{b} \\

& \mathbf{T} \geq 0

\end{aligned}\end{split}\]</div>

The problem is solved with either a proximal point method <a class="reference internal" href="#references-batch-solver">[92]</a> or a Sinkhorn algorithm <a class="reference internal" href="#references-batch-solver">[2]</a>. Unlike the Sinkhorn algorithm, which assumes a regularization term, the proximal point method can solve both regularized and unregularized optimal transport problems. When <cite>method</cite> is set to ‘auto’, the function automatically selects the appropriate method based on the value of <cite>reg</cite>. if <cite>reg</cite> is None or 0, the proximal point method is used. If <cite>reg</cite> is greater than 0, the Sinkhorn algorithm is used.

<dt class="field-odd">Parameters:</dt>

<li>M (array-like, shape (B, ns, nt)) – Cost matrix</li>

<li>reg (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>) – Regularization parameter. Default is None.</li>

<li>a (array-like, shape (B, ns)) – Source distribution (optional). If None, uniform distribution is used.</li>

<li>b (array-like, shape (B, nt)) – Target distribution (optional). If None, uniform distribution is used.</li>

<li>method (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)">str</a>) – Method to use, either ‘auto’, ‘proximal’, ‘log_sinkhorn’ or ‘sinkhorn’. Default is ‘auto’.</li>

<li>inner_iter (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)">int</a>) – Number of inner Bregman iterations for the proximal method. Default is 1.</li>

<li>inner_reg (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>) – Regularization parameter for the inner Bregman iterations in the proximal method. Default is 1e-3.</li>

<li>reg_type (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)">str</a>, optional) – Type of regularization \(R\) either “KL”, or “entropy”. Default is “entropy”.</li>

<li>grad (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)">str</a>, optional) – Type of gradient computation, either ‘detach’, ‘autodiff’, ‘last_step’ or ‘envelope’.

‘detach’ does not compute the gradients.

‘autodiff’ provides gradients of all outputs (<cite>plan, value, value_linear</cite>) but with important memory cost.

‘last_step’ provides gradients of all outputs (<cite>plan, value, value_linear</cite>) only for the last method iteration, useful for memory saving.

‘envelope’ provides gradients only for <cite>value</cite>.

Default is ‘envelope’.</li>

</ul>

</dd>

<dt class="field-even">Returns:</dt>

<dd class="field-even">res – Result of the optimization problem. The information can be obtained as follows:

<li>res.plan : OT plan \(\mathbf{T}\)</li>

<li>res.potentials : OT dual potentials</li>

<li>res.value : Optimal value of the optimization problem</li>

<li>res.value_linear : Linear OT loss with the optimal OT plan</li>

</ul>

See <a class="reference internal" href="ot.utils.html#id56" title="ot.utils.OTResult"><code class="xref any py py-class docutils literal notranslate">OTResult</code></a> for more information.

</dd>

<dt class="field-odd">Return type:</dt>

<dd class="field-odd"><a class="reference internal" href="ot.utils.html#id56" title="ot.utils.OTResult">OTResult</a>()

</dd>

</dl>

Examples

>>> X = np.random.randn(5, 10, 3) # 5 batches of 10 samples in 3D

>>> Y = np.random.randn(5, 15, 3) # 5 batches of 15 samples in 3D

>>> M = dist_batch(X, Y, metric="euclidean") # Compute cost matrices

>>> p_result = solve_batch(M) # Uses proximal method

>>> reg = 0.1

>>> s_result = solve_batch(M, reg, method="log_sinkhorn") # Uses Sinkhorn method

>>> s_result.plan.shape # Optimal transport plans for each batch

(5, 10, 15)

>>> s_result.value.shape # Optimal transport values for each batch

(5,)

</pre></div>

</div>

See also

ot.batch.dist_batch : batched cost matrix computation for computing M.

ot.solve : non-batched version of the solve_batch function.

[92]

Xie, Y., Wang, X., Wang, R., & Zha, H. (2020, August).

</aside>

A fast proximal point method for computing exact wasserstein distance.

In Uncertainty in artificial intelligence (pp. 433-453). PMLR.

[2]

<li>Cuturi, Sinkhorn Distances : Lightspeed Computation</li>

</ol>

</aside>

of Optimal Transport, Advances in Neural Information Processing

Systems (NIPS) 26, 2013

</div>

</dd></dl>

ot.batch.solve_gromov_batch(Ca, Cb, reg=0.01, a=None, b=None, loss='sqeuclidean', symmetric=None, M=None, alpha=None, T_init=None, max_iter=50, tol=1e-05, max_iter_inner=50, tol_inner=1e-05, grad='envelope', logits=None)<a class="reference internal" href="../_modules/ot/batch/_quadratic.html#solve_gromov_batch">[source]</a><a class="headerlink" href="#ot.batch.solve_gromov_batch" title="Link to this definition"></a></dt>

<dd>Solves a batch of Gromov-Wasserstein optimal transport problems using proximal gradient [12, 81].

For each problem in the batch, solves:

\[\begin{split}\begin{aligned}

\min_{\mathbf{T} \geq 0} \quad & \sum_{i,j,k,l} L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j} \mathbf{T}_{k,l} \\

\text{s.t.} \quad & \mathbf{T} \mathbf{1} = \mathbf{a} \\

& \mathbf{T}^T \mathbf{1} = \mathbf{b} \\

& \mathbf{T} \geq 0

\end{aligned}\end{split}\]</div>

If \(\mathbf{M}\) and \(\alpha\) are given, solves the more general fused Gromov-Wasserstein problem:

\[\begin{split}\begin{aligned}

\min_{\mathbf{T} \geq 0} \quad & (1-\alpha) \sum_{i,j} M_{i,j} \mathbf{T}_{i,j} + \alpha \sum_{i,j,k,l} L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j} \mathbf{T}_{k,l} \\

\text{s.t.} \quad & \mathbf{T} \mathbf{1} = \mathbf{a} \\

& \mathbf{T}^T \mathbf{1} = \mathbf{b} \\

& \mathbf{T} \geq 0

\end{aligned}\end{split}\]</div>

Writing the objective as \((1-\alpha) \langle \mathbf{M}, \mathbf{T} \rangle + \alpha \langle \mathcal{L} \otimes \mathbf{T}, \mathbf{T} \rangle\),

the solver uses proximal gradient descent where each iteration is:

\[\begin{split}\begin{aligned}

\mathbf{T}_{k+1} = \mathop{\arg \min}_{\mathbf{T} \geq 0} \quad & \langle \mathbf{M}_k, \mathbf{T} \rangle + \epsilon \, \text{KL}(\mathbf{T} \| \mathbf{T}_k) \\

\text{where} \quad & \mathbf{M}_k = (1 - \alpha) \mathbf{M} + 2 \alpha \mathcal{L} \otimes \mathbf{T}_k

\end{aligned}\end{split}\]</div>

This can be rewritten as:

\[\begin{aligned}

\mathbf{T}_{k+1} = \mathop{\arg \min}_{\mathbf{T} \geq 0} \quad & \langle \mathbf{M}_k - \epsilon \log(\mathbf{T}_k), \mathbf{T} \rangle - \epsilon H(\mathbf{T})

\end{aligned}\]</div>

where \(H(\mathbf{T})\) is the entropy of \(\mathbf{T}\). Thus each iteration can be solved using the Bregman projection solver implemented in <cite>bregman_log_projection_batch</cite>.

Note that the inner optimization problem does not need to be solved exactly. In practice it sufficient to set <cite>max_iter_inner</cite> to a low value (e.g. 20) and/or <cite>tol_inner</cite> to a high value (e.g. 1e-2).

<dt class="field-odd">Parameters:</dt>

<li>Ca (array-like, shape (B, n, n, d) or (B, n, n)) – Samples affinity matrices from source distribution</li>

<li>Cb (array-like, shape (B, n, n, d) or (B, n, n)) – Samples affinity matrices from target distribution</li>

<li>a (array-like, shape (B, n), optional) – Marginal distribution of the source samples. If None, uniform distribution is used.</li>

<li>b (array-like, shape (B, m), optional) – Marginal distribution of the target samples. If None, uniform distribution is used.</li>

<li>loss (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)">str</a>, optional) – Type of loss function, can be ‘sqeuclidean’ or ‘kl’ or a QuadraticMetric instance.</li>

<li>symmetric (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)">bool</a>, optional) – Either Ca and Cb are to be assumed symmetric or not.

If let to its default None value, a symmetry test will be conducted.

Else if set to True (resp. False), Ca and Cb will be assumed symmetric (resp. asymmetric).</li>

<li>M (array-like, shape (dim_a, dim_b), optional) – Linear cost matrix for Fused Gromov-Wasserstein (default is None).</li>

((1-alpha)*Wass) in the Fused Gromov-Wasserstein problem. Not used for

Gromov problem (when M is not provided). By default <code class="docutils literal notranslate">alpha=None</code>

<li>epsilon (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>, optional) – Regularization parameter for proximal gradient descent. Default is 1e-2.</li>

<li>T_init (array-like, shape (B, n, m), optional) – Initial transport plan. If None, it is initialized to uniform distribution.</li>

<li>max_iter_inner (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)">int</a>, optional) – Maximum number of iterations for the inner Bregman projection. Default is 50.</li>

<li>tol_inner (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>, optional) – Tolerance for convergence of the inner Bregman projection. Default is 1e-5.</li>

<li>grad (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)">str</a>, optional) – Type of gradient computation, either or ‘autodiff’, ‘envelope’ or ‘detach’. ‘autodiff’ provides gradients wrt all

outputs (<cite>plan, value, value_linear</cite>) but with important memory cost.

‘envelope’ provides gradients only for (<cite>value, value_linear</cite>)`. <cite>detach`</cite> is the fastest option but

provides no gradients. Default is ‘detach’.</li>

<li>assume_inner_convergence (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)">bool</a>, optional) – If True, assumes that the inner Bregman projection always converged i.e. the transport plan satisfies the marginal constraints.

This enables faster computations of the tensor product but might results in inaccurate results (e.g. negative values of the loss).

Default is True.</li>

</ul>

</dd>

<dt class="field-even">Returns:</dt>

<dd class="field-even">res – Result of the optimization problem. The information can be obtained as follows:

<li>res.plan : OT plan \(\mathbf{T}\)</li>

<li>res.potentials : OT dual potentials</li>

<li>res.value : Optimal value of the optimization problem</li>

<li>res.value_linear : Linear OT loss with the optimal OT plan</li>

<li>res.value_quad : Quadratic OT loss with the optimal OT plan</li>

</ul>

</dd>

<dt class="field-odd">Return type:</dt>

<dd class="field-odd"><a class="reference internal" href="ot.utils.html#id56" title="ot.utils.OTResult">OTResult</a>()

</dd>

</dl>

See also

</dd>

<dt><a class="reference internal" href="../all.html#ot.solve_gromov" title="ot.solve_gromov"><code class="xref py py-obj docutils literal notranslate">ot.solve_gromov</code></a></dt><dd>Non-batched solver for Gromov-Wasserstein. Note that the non-batched solver uses a different algorithm (conditional gradient) and might not give the same results.

</dd>

</dl>

</div>

References

[12]

Gabriel Peyré, Marco Cuturi, and Justin Solomon,

“Gromov-Wasserstein averaging of kernel and distance matrices.”

International Conference on Machine Learning (ICML). 2016.

</aside>

[81]

Xu, H., Luo, D., & Carin, L. (2019). “Scalable Gromov-Wasserstein learning for graph partitioning and matching.”

Advances in neural information processing systems (NeurIPS). 2019.

</aside>

</dd></dl>

ot.batch.solve_sample_batch(X_a, X_b, reg=None, a=None, b=None, metric='sqeuclidean', p=2, max_iter=1000, tol=1e-05, method='auto', inner_iter=1, inner_reg=0.001, reg_type='entropy', grad='envelope')<a class="reference internal" href="../_modules/ot/batch/_linear.html#solve_sample_batch">[source]</a><a class="headerlink" href="#ot.batch.solve_sample_batch" title="Link to this definition"></a></dt>

The problem is solved with either a proximal point method <a class="reference internal" href="#references-batch-solver">[91]</a> or a Sinkhorn algorithm <a class="reference internal" href="#references-batch-solver">[2]</a>. Unlike the Sinkhorn algorithm, which assumes a regularization term, the proximal point method can solve both regularized and unregularized optimal transport problems. When <cite>method</cite> is set to ‘auto’, the function automatically selects the appropriate method based on the value of <cite>reg</cite>. if <cite>reg</cite> is None or 0, the proximal point method is used. If <cite>reg</cite> is greater than 0, the Sinkhorn algorithm is used.

<dt class="field-odd">Parameters:</dt>

<li>X_a (array-like, shape (B, ns, d)) – Samples from source distribution</li>

<li>X_b (array-like, shape (B, nt, d)) – Samples from target distribution</li>

<li>inner_iter (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)">int</a>) – Number of inner Bregman iterations for the proximal method. Default is 1.</li>

<li>inner_reg (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>) – Regularization parameter for the inner Bregman iterations in the proximal method. Default is 1e-3.</li>

<li>grad (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)">str</a>, optional) – Type of gradient computation, either ‘detach’, ‘autodiff’, ‘last_step’ or ‘envelope’.

‘detach’ does not compute the gradients.

‘autodiff’ provides gradients of all outputs (<cite>plan, value, value_linear</cite>) but with important memory cost.

‘last_step’ provides gradients of all outputs (<cite>plan, value, value_linear</cite>) only for the last method iteration, useful for memory saving.

‘envelope’ provides gradients only for <cite>value</cite>.

Default is ‘envelope’.</li>

</ul>

</dd>

<dt class="field-even">Returns:</dt>

<dd class="field-even">res – Result of the optimization problem. The information can be obtained as follows:

<li>res.plan : OT plan \(\mathbf{T}\)</li>

<li>res.potentials : OT dual potentials</li>

<li>res.value : Optimal value of the optimization problem</li>

<li>res.value_linear : Linear OT loss with the optimal OT plan</li>

</ul>

</dd>

<dt class="field-odd">Return type:</dt>

<dd class="field-odd"><a class="reference internal" href="ot.utils.html#id56" title="ot.utils.OTResult">OTResult</a>()

</dd>

</dl>

See also

</dd>

</dl>

</div>

</dd></dl>

ot.batch.tensor_batch(a, b, Ca, Cb, symmetric=True, nx=None, loss='sqeuclidean', logits=None)<a class="reference internal" href="../_modules/ot/batch/_quadratic.html#tensor_batch">[source]</a><a class="headerlink" href="#ot.batch.tensor_batch" title="Link to this definition"></a></dt>

<dd>Compute the Gromov-Wasserstein cost tensor for a batch of problems.

The Gromov-Wasserstein distance can be expressed as:

\[\text{GW}(\mathbf{T}, \mathbf{C}_1, \mathbf{C}_2) = \sum_{ijkl} T_{ik} T_{jl} \ell(C_{1,ij}, C_{2,kl}) = \langle \mathcal{L} \times \mathbf{T}, \mathbf{T} \rangle\]</div>

where \(\mathcal{L}\) is a 4D tensor with elements \(\mathcal{L}[i,j,k,l] = \ell(C_{1,ij}, C_{2,kl})\).

For loss functions of the form \(\ell(a,b) = f_1(a) + f_2(b) - \langle h_1(a), h_2(b) \rangle\),

the tensor product \(\mathcal{L} \times \mathbf{T}\) can be computed efficiently without explicitly computing \(\mathcal{L}\) [12].

This function precomputes all matrices that implicitly define <cite>mathcal{L}</cite> for various loss functions.

<dt class="field-odd">Parameters:</dt>

<li>a (array-like, shape (B, n)) – Source distributions for each problem in the batch.</li>

<li>b (array-like, shape (B, m)) – Target distributions for each problem in the batch.</li>

<li>nx (backend object, optional) – Numerical backend to use for computations. If None, the default backend is used.</li>

Default is ‘sqeuclidean’.</li>

If True, inputs are treated as logits. Default is None.</li>

</ul>

</dd>

<dt class="field-even">Returns:</dt>

<dd class="field-even">Dictionary containing:

- constC : array-like, shape (B, n, m)

<div>Constant term in the tensor product.

</div></blockquote>

<li>hCa : array-like, shape (B, n, n, d) or (B, n, n)</li>

<li>hCb : array-like, shape (B, m, m, d) or (B, m, m)</li>

<li>fCa : array-like, shape (B, n, n)</li>

<li>fCb : array-like, shape (B, m, m)</li>

</ul>

</dd>

<dt class="field-odd">Return type:</dt>

</dd>

</dl>

<h2>Supported loss functions:<a class="headerlink" href="#supported-loss-functions" title="Link to this heading"></a></h2>

Squared Euclidean loss:

\[\ell(a, b) = \|a - b\|_2^2 = \sum_i (a_i - b_i)^2\]</div>

KL divergence:

\[\ell(a, b) = \sum_i a_i \log\left(\frac{a_i}{b_i}\right)\]</div>

If <code class="docutils literal notranslate">logits=True</code>, the entries of Ca are treated as logits (unnormalized log probabilities)

and the loss becomes:

\[\ell(x, y) = \sum_i y_i (\log(y_i) - x_i)\]</div>

Examples

>>> # Create batch of cost matrices

>>> L = tensor_batch(a, b, Ca, Cb, loss='sqeuclidean')

</pre></div>

</div>

References

[12]

Gabriel Peyré, Marco Cuturi, and Justin Solomon,

“Gromov-Wasserstein averaging of kernel and distance matrices.”

International Conference on Machine Learning (ICML). 2016.

</aside>

</section>

</dd></dl>

</section>

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