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<!DOCTYPE html>

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<title>ot.batch — POT Python Optimal Transport 0.9.8.dev 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/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional) – Maximum number of iterations.</li>

<li>tol (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">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/builtins/stdtypes.html#str" title="(in Python v3.15)">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/builtins/stdtypes.html#str" title="(in Python v3.15)">str</a>, optional) – ‘sqeuclidean’, ‘euclidean’, ‘minkowski’ or ‘kl’</li>

<li>p (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">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/builtins/functions.html#float" title="(in Python v3.15)">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/builtins/functions.html#float" title="(in Python v3.15)">float</a>, array-like or <a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#list" title="(in Python v3.15)">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/builtins/stdtypes.html#list" title="(in Python v3.15)">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/builtins/stdtypes.html#str" title="(in Python v3.15)">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/builtins/functions.html#bool" title="(in Python v3.15)">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/builtins/functions.html#bool" title="(in Python v3.15)">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/builtins/functions.html#bool" title="(in Python v3.15)">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/builtins/functions.html#bool" title="(in Python v3.15)">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/builtins/functions.html#int" title="(in Python v3.15)">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/builtins/functions.html#float" title="(in Python v3.15)">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/builtins/functions.html#float" title="(in Python v3.15)">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/builtins/stdtypes.html#str" title="(in Python v3.15)">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/builtins/functions.html#int" title="(in Python v3.15)">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/builtins/functions.html#float" title="(in Python v3.15)">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/builtins/stdtypes.html#str" title="(in Python v3.15)">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/builtins/stdtypes.html#str" title="(in Python v3.15)">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/builtins/stdtypes.html#str" title="(in Python v3.15)">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/builtins/functions.html#bool" title="(in Python v3.15)">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>reg (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">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/builtins/functions.html#int" title="(in Python v3.15)">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/builtins/functions.html#float" title="(in Python v3.15)">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/builtins/stdtypes.html#str" title="(in Python v3.15)">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 ‘envelope’.</li>

<li>logits (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#bool" title="(in Python v3.15)">bool</a>, optional) – For KL loss, whether Ca and Cb are logits (unnormalized log

probabilities). Default is None.</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/builtins/functions.html#int" title="(in Python v3.15)">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/builtins/functions.html#float" title="(in Python v3.15)">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/builtins/stdtypes.html#str" title="(in Python v3.15)">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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