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<!DOCTYPE html>

<head>

<title>ot.bregman — 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.bregman.barycenter"><code class="docutils literal notranslate">barycenter()</code></a></li>

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

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

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

<li class="toctree-l3"><a class="reference internal" href="#ot.bregman.convolutional_barycenter2d"><code class="docutils literal notranslate">convolutional_barycenter2d()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#ot.bregman.convolutional_barycenter2d_debiased"><code class="docutils literal notranslate">convolutional_barycenter2d_debiased()</code></a></li>

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

<li class="toctree-l3"><a class="reference internal" href="#ot.bregman.empirical_sinkhorn2"><code class="docutils literal notranslate">empirical_sinkhorn2()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#ot.bregman.empirical_sinkhorn2_geomloss"><code class="docutils literal notranslate">empirical_sinkhorn2_geomloss()</code></a></li>

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

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

<li class="toctree-l3"><a class="reference internal" href="#ot.bregman.empirical_sinkhorn_nystroem2"><code class="docutils literal notranslate">empirical_sinkhorn_nystroem2()</code></a></li>

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

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

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

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

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

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

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

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

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

<li class="toctree-l3"><a class="reference internal" href="#ot.bregman.sinkhorn2"><code class="docutils literal notranslate">sinkhorn2()</code></a></li>

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

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

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

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

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

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

Solvers related to Bregman projections for entropic regularized OT

ot.bregman.barycenter(A, M, reg, weights=None, method='sinkhorn', numItermax=10000, stopThr=0.0001, verbose=False, log=False, warn=True, **kwargs)<a class="reference internal" href="../_modules/ot/bregman/_barycenter.html#barycenter">[source]</a><a class="headerlink" href="#ot.bregman.barycenter" title="Link to this definition"></a></dt>

<dd>Compute the entropic regularized wasserstein barycenter of distributions \(\mathbf{A}\)

<div>The function solves the following optimization problem:

</div></blockquote>

\[\mathbf{a} = \mathop{\arg \min}_\mathbf{a} \quad \sum_i W_{reg}(\mathbf{a},\mathbf{a}_i)\]</div>

where :

<li>\(W_{reg}(\cdot,\cdot)\) is the entropic regularized Wasserstein

distance (see <a class="reference internal" href="#ot.bregman.sinkhorn" title="ot.bregman.sinkhorn"><code class="xref py py-func docutils literal notranslate">ot.bregman.sinkhorn()</code></a>)

if <cite>method</cite> is <cite>sinkhorn</cite> or <cite>sinkhorn_stabilized</cite> or <cite>sinkhorn_log</cite>.</li>

<li>\(\mathbf{a}_i\) are training distributions in the columns of matrix

\(\mathbf{A}\)</li>

<li><cite>reg</cite> and \(\mathbf{M}\) are respectively the regularization term and

the cost matrix for OT</li>

</ul>

The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling

algorithm as proposed in <a class="reference internal" href="#references-barycenter">[3]</a>

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

<li>A (array-like, shape (dim, n_hists)) – <cite>n_hists</cite> training distributions \(\mathbf{a}_i\) of size <cite>dim</cite></li>

<li>M (array-like, shape (dim, dim)) – loss matrix for OT</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 term > 0</li>

<li>method (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)">str</a> (optional)) – method used for the solver either ‘sinkhorn’ or ‘sinkhorn_stabilized’ or ‘sinkhorn_log’</li>

<li>weights (array-like, shape (n_hists,)) – Weights of each histogram \(\mathbf{a}_i\) on the simplex (barycentric coordinates)</li>

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

<li>stopThr (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>, optional) – Stop threshold on error (>0)</li>

<li>verbose (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)">bool</a>, optional) – Print information along iterations</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) – record log if True</li>

<li>warn (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)">bool</a>, optional) – if True, raises a warning if the algorithm doesn’t convergence.</li>

</ul>

</dd>

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

<li>a ((dim,) array-like) – Wasserstein barycenter</li>

<li>log (dict) – log dictionary return only if log==True in parameters</li>

</ul>

</dd>

</dl>

References

[3]

Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015).

Iterative Bregman projections for regularized transportation problems.

SIAM Journal on Scientific Computing, 37(2), A1111-A1138.

</aside>

</dd></dl>

ot.bregman.barycenter_debiased(A, M, reg, weights=None, method='sinkhorn', numItermax=10000, stopThr=0.0001, verbose=False, log=False, warn=True, **kwargs)<a class="reference internal" href="../_modules/ot/bregman/_barycenter.html#barycenter_debiased">[source]</a><a class="headerlink" href="#ot.bregman.barycenter_debiased" title="Link to this definition"></a></dt>

<dd>Compute the debiased Sinkhorn barycenter of distributions A

<div>The function solves the following optimization problem:

</div></blockquote>

\[\mathbf{a} = \mathop{\arg \min}_\mathbf{a} \quad \sum_i S_{reg}(\mathbf{a},\mathbf{a}_i)\]</div>

where :

<li>\(S_{reg}(\cdot,\cdot)\) is the debiased Sinkhorn divergence

(see <a class="reference internal" href="#ot.bregman.empirical_sinkhorn_divergence" title="ot.bregman.empirical_sinkhorn_divergence"><code class="xref py py-func docutils literal notranslate">ot.bregman.empirical_sinkhorn_divergence()</code></a>)</li>

<li>\(\mathbf{a}_i\) are training distributions in the columns of matrix

\(\mathbf{A}\)</li>

<li><cite>reg</cite> and \(\mathbf{M}\) are respectively the regularization term and

the cost matrix for OT</li>

</ul>

The algorithm used for solving the problem is the debiased Sinkhorn

algorithm as proposed in <a class="reference internal" href="#references-barycenter-debiased">[37]</a>

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

<li>M (array-like, shape (dim, dim)) – loss matrix for OT</li>

</ul>

</dd>

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

<li>a ((dim,) array-like) – Wasserstein barycenter</li>

<li>log (dict) – log dictionary return only if log==True in parameters</li>

</ul>

</dd>

</dl>

References

[37]

Janati, H., Cuturi, M., Gramfort, A. Proceedings of the 37th International

Conference on Machine Learning, PMLR 119:4692-4701, 2020

</aside>

</dd></dl>

ot.bregman.barycenter_sinkhorn(A, M, reg, weights=None, numItermax=1000, stopThr=0.0001, verbose=False, log=False, warn=True)<a class="reference internal" href="../_modules/ot/bregman/_barycenter.html#barycenter_sinkhorn">[source]</a><a class="headerlink" href="#ot.bregman.barycenter_sinkhorn" title="Link to this definition"></a></dt>

<dd>Compute the entropic regularized wasserstein barycenter of distributions \(\mathbf{A}\)

<div>The function solves the following optimization problem:

</div></blockquote>

\[\mathbf{a} = \mathop{\arg \min}_\mathbf{a} \quad \sum_i W_{reg}(\mathbf{a},\mathbf{a}_i)\]</div>

where :

<li>\(W_{reg}(\cdot,\cdot)\) is the entropic regularized Wasserstein distance

(see <a class="reference internal" href="#ot.bregman.sinkhorn" title="ot.bregman.sinkhorn"><code class="xref py py-func docutils literal notranslate">ot.bregman.sinkhorn()</code></a>)</li>

<li>\(\mathbf{a}_i\) are training distributions in the columns of matrix

\(\mathbf{A}\)</li>

<li><cite>reg</cite> and \(\mathbf{M}\) are respectively the regularization term and

the cost matrix for OT</li>

</ul>

The algorithm used for solving the problem is the Sinkhorn-Knopp matrix

scaling algorithm as proposed in <a class="reference internal" href="#references-barycenter-sinkhorn">[3]</a>.

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

<li>M (array-like, shape (dim, dim)) – loss matrix for OT</li>

</ul>

</dd>

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

<li>a ((dim,) array-like) – Wasserstein barycenter</li>

<li>log (dict) – log dictionary return only if log==True in parameters</li>

</ul>

</dd>

</dl>

References

[3]

Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015).

</aside>

Iterative Bregman projections for regularized transportation problems.

SIAM Journal on Scientific Computing, 37(2), A1111-A1138.

</dd></dl>

ot.bregman.barycenter_stabilized(A, M, reg, tau=10000000000.0, weights=None, numItermax=1000, stopThr=0.0001, verbose=False, log=False, warn=True)<a class="reference internal" href="../_modules/ot/bregman/_barycenter.html#barycenter_stabilized">[source]</a><a class="headerlink" href="#ot.bregman.barycenter_stabilized" title="Link to this definition"></a></dt>

<dd>Compute the entropic regularized wasserstein barycenter of distributions \(\mathbf{A}\) with stabilization.

<div>The function solves the following optimization problem:

</div></blockquote>

\[\mathbf{a} = \mathop{\arg \min}_\mathbf{a} \quad \sum_i W_{reg}(\mathbf{a},\mathbf{a}_i)\]</div>

where :

<li>\(W_{reg}(\cdot,\cdot)\) is the entropic regularized Wasserstein

<li>\(\mathbf{a}_i\) are training distributions in the columns of matrix

\(\mathbf{A}\)</li>

<li><cite>reg</cite> and \(\mathbf{M}\) are respectively the regularization term and

the cost matrix for OT</li>

</ul>

The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling

algorithm as proposed in <a class="reference internal" href="#references-barycenter-stabilized">[3]</a>

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

<li>M (array-like, shape (dim, dim)) – loss matrix for OT</li>

<li>tau (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>) – threshold for max value in \(\mathbf{u}\) or \(\mathbf{v}\)

for log scaling</li>

</ul>

</dd>

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

<li>a ((dim,) array-like) – Wasserstein barycenter</li>

<li>log (dict) – log dictionary return only if log==True in parameters</li>

</ul>

</dd>

</dl>

References

[3]

Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015).

Iterative Bregman projections for regularized transportation problems.

SIAM Journal on Scientific Computing, 37(2), A1111-A1138.

</aside>

</dd></dl>

ot.bregman.convolutional_barycenter2d(A, reg, weights=None, method='sinkhorn', numItermax=10000, stopThr=0.0001, verbose=False, log=False, warn=True, **kwargs)<a class="reference internal" href="../_modules/ot/bregman/_convolutional.html#convolutional_barycenter2d">[source]</a><a class="headerlink" href="#ot.bregman.convolutional_barycenter2d" title="Link to this definition"></a></dt>

<dd>Compute the entropic regularized wasserstein barycenter of distributions \(\mathbf{A}\)

where \(\mathbf{A}\) is a collection of 2D images.

<div>The function solves the following optimization problem:

</div></blockquote>

\[\mathbf{a} = \mathop{\arg \min}_\mathbf{a} \quad \sum_i W_{reg}(\mathbf{a},\mathbf{a}_i)\]</div>

where :

<li>\(W_{reg}(\cdot,\cdot)\) is the entropic regularized Wasserstein

<li>\(\mathbf{a}_i\) are training distributions (2D images) in the mast two dimensions

of matrix \(\mathbf{A}\)</li>

<li><cite>reg</cite> is the regularization strength scalar value</li>

</ul>

The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm

as proposed in <a class="reference internal" href="#references-convolutional-barycenter-2d">[21]</a>

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

<li>A (array-like, shape (n_hists, width, height)) – <cite>n</cite> distributions (2D images) of size <cite>width</cite> x <cite>height</cite></li>

<li>weights (array-like, shape (n_hists,)) – Weights of each image on the simplex (barycentric coordinates)</li>

<li>method (string, optional) – method used for the solver either ‘sinkhorn’ or ‘sinkhorn_log’</li>

<li>stabThr (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>, optional) – Stabilization threshold to avoid numerical precision issue</li>

</ul>

</dd>

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

<li>a (array-like, shape (width, height)) – 2D Wasserstein barycenter</li>

<li>log (dict) – log dictionary return only if log==True in parameters</li>

</ul>

</dd>

</dl>

References

[21]

Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher,

A., Nguyen, A. & Guibas, L. (2015). Convolutional wasserstein distances:

Efficient optimal transportation on geometric domains. ACM Transactions

on Graphics (TOG), 34(4), 66

</aside>

[37]

Janati, H., Cuturi, M., Gramfort, A. Proceedings of the 37th

International Conference on Machine Learning, PMLR 119:4692-4701, 2020

</aside>

</dd></dl>

ot.bregman.convolutional_barycenter2d_debiased(A, reg, weights=None, method='sinkhorn', numItermax=10000, stopThr=0.001, verbose=False, log=False, warn=True, **kwargs)<a class="reference internal" href="../_modules/ot/bregman/_convolutional.html#convolutional_barycenter2d_debiased">[source]</a><a class="headerlink" href="#ot.bregman.convolutional_barycenter2d_debiased" title="Link to this definition"></a></dt>

<dd>Compute the debiased sinkhorn barycenter of distributions \(\mathbf{A}\)

where \(\mathbf{A}\) is a collection of 2D images.

<div>The function solves the following optimization problem:

</div></blockquote>

\[\mathbf{a} = \mathop{\arg \min}_\mathbf{a} \quad \sum_i S_{reg}(\mathbf{a},\mathbf{a}_i)\]</div>

where :

<li>\(S_{reg}(\cdot,\cdot)\) is the debiased entropic regularized Wasserstein

distance (see <a class="reference internal" href="#ot.bregman.barycenter_debiased" title="ot.bregman.barycenter_debiased"><code class="xref py py-func docutils literal notranslate">ot.bregman.barycenter_debiased()</code></a>)</li>

<li>\(\mathbf{a}_i\) are training distributions (2D images) in the mast two

dimensions of matrix \(\mathbf{A}\)</li>

<li><cite>reg</cite> is the regularization strength scalar value</li>

</ul>

The algorithm used for solving the problem is the debiased Sinkhorn scaling

algorithm as proposed in <a class="reference internal" href="#references-convolutional-barycenter2d-debiased">[37]</a>

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

<li>method (string, optional) – method used for the solver either ‘sinkhorn’ or ‘sinkhorn_log’</li>

</ul>

</dd>

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

<li>a (array-like, shape (width, height)) – 2D Wasserstein barycenter</li>

<li>log (dict) – log dictionary return only if log==True in parameters</li>

</ul>

</dd>

</dl>

References

[37]

Janati, H., Cuturi, M., Gramfort, A. Proceedings of the 37th International

Conference on Machine Learning, PMLR 119:4692-4701, 2020

</aside>

</dd></dl>

ot.bregman.empirical_sinkhorn(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', numIterMax=10000, stopThr=1e-09, isLazy=False, batchSize=100, verbose=False, log=False, warn=True, warmstart=None, **kwargs)<a class="reference internal" href="../_modules/ot/bregman/_empirical.html#empirical_sinkhorn">[source]</a><a class="headerlink" href="#ot.bregman.empirical_sinkhorn" title="Link to this definition"></a></dt>

<dd>Solve the entropic regularization optimal transport problem and return the

OT matrix from empirical data

The function solves the following optimization problem:

\[ \begin{align}\begin{aligned}\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg} \cdot\Omega(\gamma)\\s.t. \ \gamma \mathbf{1} &= \mathbf{a}\\ \gamma^T \mathbf{1} &= \mathbf{b}\\ \gamma &\geq 0\end{aligned}\end{align} \]</div>

where :

<li>\(\mathbf{M}\) is the (<cite>n_samples_a</cite>, <cite>n_samples_b</cite>) metric cost matrix</li>

<li>\(\Omega\) is the entropic regularization term

\(\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})\)</li>

<li>\(\mathbf{a}\) and \(\mathbf{b}\) are source and target weights (sum to 1)</li>

</ul>

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

<li>X_s (array-like, shape (n_samples_a, dim)) – samples in the source domain</li>

<li>X_t (array-like, shape (n_samples_b, dim)) – samples in the target domain</li>

<li>a (array-like, shape (n_samples_a,)) – samples weights in the source domain</li>

<li>b (array-like, shape (n_samples_b,)) – samples weights in the target domain</li>

<li>isLazy (boolean, optional) – If True, then only calculate the cost matrix by block and return

the dual potentials only (to save memory). If False, calculate full

cost matrix and return outputs of sinkhorn function.</li>

<li>batchSize (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)">int</a> or <a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#tuple" title="(in Python v3.14)">tuple</a> of 2 int, optional) – Size of the batches used to compute the sinkhorn update without memory overhead.

When a tuple is provided it sets the size of the left/right batches.</li>

<li>warmstart (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#tuple" title="(in Python v3.14)">tuple</a> of arrays, shape (dim_a, dim_b), optional) – Initialization of dual potentials. If provided, the dual potentials should be given

(that is the logarithm of the u,v sinkhorn scaling vectors)</li>

</ul>

</dd>

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

<li>gamma (array-like, shape (n_samples_a, n_samples_b)) – Regularized optimal transportation matrix for the given parameters</li>

<li>log (dict) – log dictionary return only if log==True in parameters</li>

</ul>

</dd>

</dl>

Examples

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

>>> n_samples_a = 2

>>> n_samples_b = 2

>>> reg = 0.1

>>> X_s = np.reshape(np.arange(n_samples_a, dtype=np.float64), (n_samples_a, 1))

>>> X_t = np.reshape(np.arange(0, n_samples_b, dtype=np.float64), (n_samples_b, 1))

>>> empirical_sinkhorn(X_s, X_t, reg=reg, verbose=False)

array([[4.99977301e-01, 2.26989344e-05],

[2.26989344e-05, 4.99977301e-01]])

</pre></div>

</div>

References

[2]

M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal

Transport, Advances in Neural Information Processing Systems (NIPS) 26, 2013

</aside>

[9]

Schmitzer, B. (2016). Stabilized Sparse Scaling Algorithms for

Entropy Regularized Transport Problems. arXiv preprint arXiv:1610.06519.

</aside>

[10]

Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016).

Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.

</aside>

</dd></dl>

ot.bregman.empirical_sinkhorn2(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', numIterMax=10000, stopThr=1e-09, isLazy=False, batchSize=100, verbose=False, log=False, warn=True, warmstart=None, **kwargs)<a class="reference internal" href="../_modules/ot/bregman/_empirical.html#empirical_sinkhorn2">[source]</a><a class="headerlink" href="#ot.bregman.empirical_sinkhorn2" title="Link to this definition"></a></dt>

<dd>Solve the entropic regularization optimal transport problem from empirical

data and return the OT loss

The function solves the following optimization problem:

\[ \begin{align}\begin{aligned}W = \min_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg} \cdot\Omega(\gamma)\\s.t. \ \gamma \mathbf{1} &= \mathbf{a}\\ \gamma^T \mathbf{1} &= \mathbf{b}\\ \gamma &\geq 0\end{aligned}\end{align} \]</div>

where :

<li>\(\mathbf{M}\) is the (<cite>n_samples_a</cite>, <cite>n_samples_b</cite>) metric cost matrix</li>

<li>\(\Omega\) is the entropic regularization term

\(\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})\)</li>

<li>\(\mathbf{a}\) and \(\mathbf{b}\) are source and target weights (sum to 1)</li>

</ul>

and returns \(\langle \gamma^*, \mathbf{M} \rangle_F\) (without

the entropic contribution).

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

<li>X_s (array-like, shape (n_samples_a, dim)) – samples in the source domain</li>

<li>X_t (array-like, shape (n_samples_b, dim)) – samples in the target domain</li>

<li>a (array-like, shape (n_samples_a,)) – samples weights in the source domain</li>

<li>b (array-like, shape (n_samples_b,)) – samples weights in the target domain</li>

<li>isLazy (boolean, optional) – If True, then only calculate the cost matrix by block and return

the dual potentials only (to save memory). If False, calculate

full cost matrix and return outputs of sinkhorn function.</li>

When a tuple is provided it sets the size of the left/right batches.</li>

(that is the logarithm of the u,v sinkhorn scaling vectors)</li>

</ul>

</dd>

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

<li>W ((n_hists) array-like or float) – Optimal transportation loss for the given parameters</li>

<li>log (dict) – log dictionary return only if log==True in parameters</li>

</ul>

</dd>

</dl>

Examples

>>> n_samples_a = 2

>>> n_samples_b = 2

>>> reg = 0.1

>>> b = np.full((n_samples_b, 3), 1/n_samples_b)

>>> empirical_sinkhorn2(X_s, X_t, b=b, reg=reg, verbose=False)

array([4.53978687e-05, 4.53978687e-05, 4.53978687e-05])

</pre></div>

</div>

References

[2]

M. Cuturi, Sinkhorn Distances : Lightspeed Computation

of Optimal Transport, Advances in Neural Information

Processing Systems (NIPS) 26, 2013

</aside>

[9]

Schmitzer, B. (2016). Stabilized Sparse Scaling

Algorithms for Entropy Regularized Transport Problems.

arXiv preprint arXiv:1610.06519.

</aside>

[10]

Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016).

Scaling algorithms for unbalanced transport problems.

arXiv preprint arXiv:1607.05816.

</aside>

</dd></dl>

ot.bregman.empirical_sinkhorn2_geomloss(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', scaling=0.95, verbose=False, debias=False, log=False, backend='auto')<a class="reference internal" href="../_modules/ot/bregman/_geomloss.html#empirical_sinkhorn2_geomloss">[source]</a><a class="headerlink" href="#ot.bregman.empirical_sinkhorn2_geomloss" title="Link to this definition"></a></dt>

<dd>Solve the entropic regularization optimal transport problem with geomloss

The function solves the following optimization problem:

\[ \begin{align}\begin{aligned}\gamma = arg\min_\gamma <\gamma,C>_F + reg\cdot\Omega(\gamma)\\s.t. \gamma 1 = a\\ \gamma^T 1= b\\ \gamma\geq 0\end{aligned}\end{align} \]</div>

where :

<li>\(C\) is the cost matrix such that \(C_{i,j}=d(x_i^s,x_j^t)\) and

\(d\) is a metric.</li>

<li>\(\Omega\) is the entropic regularization term

\(\Omega(\gamma)=\sum_{i,j}\gamma_{i,j}\log(\gamma_{i,j})-\gamma_{i,j}+1\)</li>

<li>\(a\) and \(b\) are source and target weights (sum to 1)</li>

</ul>

The algorithm used for solving the problem is the Sinkhorn-Knopp matrix

scaling algorithm as proposed in and computed in log space for

better stability and epsilon-scaling. The solution is computed in a lazy way

using the Geomloss <a class="footnote-reference brackets" href="#id17" id="id14" role="doc-noteref">[60]</a> and the KeOps library <a class="footnote-reference brackets" href="#id18" id="id15" role="doc-noteref">[61]</a>.

Warning

The Geomloss library is required for this function to work. Also

when setting <cite>log=True</cite>, the dual potentials are computed using autograd and

may be slow for large problems and prevent computing backward gradients. Use

the fynction <a class="reference internal" href="../all.html#ot.solve_sample" title="ot.solve_sample"><code class="xref py py-func docutils literal notranslate">ot.solve_sample()</code></a> with <cite>method=’geomloss’</cite> for better performance and gradient computation.

</div>

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

<li>X_s (array-like, shape (n_samples_a, dim)) – samples in the source domain</li>

<li>X_t (array-like, shape (n_samples_b, dim)) – samples in the target domain</li>

<li>a (array-like, shape (n_samples_a,), default=None) – samples weights in the source domain</li>

<li>b (array-like, shape (n_samples_b,), default=None) – samples weights in the target domain</li>

<li>metric (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)">str</a>, default='sqeuclidean') – Metric used for the cost matrix computation Only accepted values are

‘sqeuclidean’ and ‘euclidean’.</li>

<li>scaling (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)">float</a>, default=0.95) – Scaling parameter used for epsilon scaling. Value close to one promote

precision while value close to zero promote speed.</li>

<li>debias (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)">bool</a>, default=False) – Use the debiased version of Sinkhorn algorithm <a href="#id48">[12]_</a>.</li>

<li>backend (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)">str</a>, default='auto') – Numerical backend for geomloss. Only ‘auto’ and ‘tensorized’ ‘online’

and ‘multiscale’ are accepted values.</li>

</ul>

</dd>

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

<li>value (float) – OT value</li>

<li>log (dict) – Log dictionary return only if log==True in parameters</li>

</ul>

</dd>

</dl>

References

[<a role="doc-backlink" href="#id14">60</a>]

Feydy, J., Roussillon, P., Trouvé, A., & Gori, P. (2019). [Fast

and scalable optimal transport for brain tractograms. In Medical Image

Computing and Computer Assisted Intervention–MICCAI 2019: 22nd

International Conference, Shenzhen, China, October 13–17, 2019,

Proceedings, Part III 22 (pp. 636-644). Springer International

Publishing.

</aside>

[<a role="doc-backlink" href="#id15">61</a>]

Charlier, B., Feydy, J., Glaunes, J. A., Collin, F. D., & Durif, G.

(2021). Kernel operations on the gpu, with autodiff, without memory

overflows. The Journal of Machine Learning Research, 22(1), 3457-3462.

</aside>

</dd></dl>

ot.bregman.empirical_sinkhorn_divergence(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', numIterMax=10000, stopThr=1e-09, verbose=False, log=False, warn=True, warmstart=None, **kwargs)<a class="reference internal" href="../_modules/ot/bregman/_empirical.html#empirical_sinkhorn_divergence">[source]</a><a class="headerlink" href="#ot.bregman.empirical_sinkhorn_divergence" title="Link to this definition"></a></dt>

<dd>Compute the sinkhorn divergence loss from empirical data

The function solves the following optimization problems and return the

sinkhorn divergence \(S\):

\[ \begin{align}\begin{aligned}W &= \min_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg} \cdot\Omega(\gamma)\\W_a &= \min_{\gamma_a} \quad \langle \gamma_a, \mathbf{M_a} \rangle_F +

\mathrm{reg} \cdot\Omega(\gamma_a)\\W_b &= \min_{\gamma_b} \quad \langle \gamma_b, \mathbf{M_b} \rangle_F +

\mathrm{reg} \cdot\Omega(\gamma_b)\\S &= W - \frac{W_a + W_b}{2}\end{aligned}\end{align} \]</div>

\[ \begin{align}\begin{aligned}s.t. \ \gamma \mathbf{1} &= \mathbf{a}\\ \gamma^T \mathbf{1} &= \mathbf{b}\\ \gamma &\geq 0\\ \gamma_a \mathbf{1} &= \mathbf{a}\\ \gamma_a^T \mathbf{1} &= \mathbf{a}\\ \gamma_a &\geq 0\\ \gamma_b \mathbf{1} &= \mathbf{b}\\ \gamma_b^T \mathbf{1} &= \mathbf{b}\\ \gamma_b &\geq 0\end{aligned}\end{align} \]</div>

where :

<li>\(\mathbf{M}\) (resp. \(\mathbf{M_a}\), \(\mathbf{M_b}\))

is the (<cite>n_samples_a</cite>, <cite>n_samples_b</cite>) metric cost matrix

(resp (<cite>n_samples_a, n_samples_a</cite>) and (<cite>n_samples_b</cite>, <cite>n_samples_b</cite>))</li>

<li>\(\Omega\) is the entropic regularization term

\(\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})\)</li>

<li>\(\mathbf{a}\) and \(\mathbf{b}\) are source and target weights (sum to 1)</li>

</ul>

and returns \(\langle \gamma^*, \mathbf{M} \rangle_F -(\langle \gamma^*_a, \mathbf{M_a} \rangle_F + \langle

\gamma^*_b , \mathbf{M_b} \rangle_F)/2\).

Sinkhorn divergence as introduced in the literature. The possibility to account for the entropic contributions

will be provided in a future release.

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

<li>X_s (array-like, shape (n_samples_a, dim)) – samples in the source domain</li>

<li>X_t (array-like, shape (n_samples_b, dim)) – samples in the target domain</li>

<li>a (array-like, shape (n_samples_a,)) – samples weights in the source domain</li>

<li>b (array-like, shape (n_samples_b,)) – samples weights in the target domain</li>

(that is the logarithm of the u,v sinkhorn scaling vectors)</li>

</ul>

</dd>

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

<li>W ((1,) array-like) – Optimal transportation symmetrized loss for the given parameters</li>

<li>log (dict) – log dictionary return only if log==True in parameters</li>

</ul>

</dd>

</dl>

Examples

>>> n_samples_a = 2

>>> n_samples_b = 4

>>> reg = 0.1

>>> empirical_sinkhorn_divergence(X_s, X_t, reg)

1.499887176049052

</pre></div>

</div>

References

[23]

Aude Genevay, Gabriel Peyré, Marco Cuturi, Learning Generative

Models with Sinkhorn Divergences, Proceedings of the Twenty-First

International Conference on Artificial Intelligence and Statistics,

(AISTATS) 21, 2018

</aside>

</dd></dl>

ot.bregman.empirical_sinkhorn_nystroem(X_s, X_t, reg=1.0, anchors=50, a=None, b=None, numItermax=1000, stopThr=1e-09, verbose=False, log=False, warn=True, warmstart=None, random_state=None)<a class="reference internal" href="../_modules/ot/bregman/_empirical.html#empirical_sinkhorn_nystroem">[source]</a><a class="headerlink" href="#ot.bregman.empirical_sinkhorn_nystroem" title="Link to this definition"></a></dt>

<dd>Solves the entropic regularization optimal transport problem with Sinkhorn and Nystroem factorization [76] and returns the

OT matrix from empirical data.

Corresponds to an approximation of entropic OT (for a squared Euclidean cost) that runs in linear time.

The number of anchors controls the level of approximation (the higher, the better the approximation, but the slower the computation becomes).

Warning: with low level of regularization, the OT plan can have non-positive values.

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

<li>X_s (array-like, shape (n_samples_a, dim)) – samples in the source domain</li>

<li>X_t (array-like, shape (n_samples_b, dim)) – samples in the target domain</li>

<li>anchors (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)">int</a>, optional) – The total number of anchors sampled for the Nystroem approximation (anchors/2 in each distribution), default 50.</li>

<li>a (array-like, shape (n_samples_a,)) – samples weights in the source domain</li>

<li>b (array-like, shape (n_samples_b,)) – samples weights in the target domain</li>

(that is the logarithm of the u,v sinkhorn scaling vectors)</li>

<li>random_state (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)">int</a>, optional) – The random state for sampling the components in each distribution.</li>

</ul>

</dd>

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

<li>gamma (LazyTensor) – OT plan as lazy tensor.</li>

<li>log (dict) – log dictionary return only if log==True in parameters</li>

</ul>

</dd>

</dl>

Examples

>>> n_samples_a = 2

>>> n_samples_b = 4

>>> reg = 0.1

>>> anchors = 3

>>> empirical_sinkhorn_nystroem(X_s, X_t, reg, anchors, random_state=42)[:]

array([[0.125, 0.125, 0.125, 0.125],

[0.125, 0.125, 0.125, 0.125]])

</pre></div>

</div>

References

[80]

Massively scalable Sinkhorn distances via the Nyström method, Jason Altschuler, Francis Bach, Alessandro Rudi, Jonathan Niles-Weed, NeurIPS 2019.

</aside>

</dd></dl>

ot.bregman.empirical_sinkhorn_nystroem2(X_s, X_t, reg=1.0, anchors=50, a=None, b=None, numItermax=1000, stopThr=1e-09, verbose=False, log=False, warn=True, warmstart=None, random_state=None)<a class="reference internal" href="../_modules/ot/bregman/_empirical.html#empirical_sinkhorn_nystroem2">[source]</a><a class="headerlink" href="#ot.bregman.empirical_sinkhorn_nystroem2" title="Link to this definition"></a></dt>

<dd>Solves the entropic regularization optimal transport problem with Sinkhorn and Nystroem factorization [76] and returns the

OT loss from empirical data.

Corresponds to an approximation of entropic OT (for a squared Euclidean cost) that runs in linear time.

The number of anchors controls the level of approximation (the higher, the better the approximation, but the slower the computation becomes).

Warning: with low level of regularization, the OT plan can have non-positive values.

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

<li>X_s (array-like, shape (n_samples_a, dim)) – samples in the source domain</li>

<li>X_t (array-like, shape (n_samples_b, dim)) – samples in the target domain</li>

<li>a (array-like, shape (n_samples_a,)) – samples weights in the source domain</li>

<li>b (array-like, shape (n_samples_b,)) – samples weights in the target domain</li>

(that is the logarithm of the u,v sinkhorn scaling vectors)</li>

</ul>

</dd>

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

<li>W (float) – Optimal transportation loss for the given parameters</li>

<li>log (dict) – log dictionary return only if log==True in parameters</li>

</ul>

</dd>

</dl>

Examples

>>> n_samples_a = 2

>>> n_samples_b = 4

>>> reg = 0.1

>>> anchors = 3

>>> empirical_sinkhorn_nystroem2(X_s, X_t, reg, anchors, random_state=42)

2.5

</pre></div>

</div>

References

[80]

Massively scalable Sinkhorn distances via the Nyström method, Jason Altschuler, Francis Bach, Alessandro Rudi, Jonathan Niles-Weed, NeurIPS 2019.

</aside>

</dd></dl>

ot.bregman.free_support_sinkhorn_barycenter(measures_locations, measures_weights, X_init, reg, b=None, weights=None, numItermax=100, numInnerItermax=1000, stopThr=1e-07, verbose=False, log=None, **kwargs)<a class="reference internal" href="../_modules/ot/bregman/_barycenter.html#free_support_sinkhorn_barycenter">[source]</a><a class="headerlink" href="#ot.bregman.free_support_sinkhorn_barycenter" title="Link to this definition"></a></dt>

<dd>Solves the free support (locations of the barycenters are optimized, not the weights) regularized Wasserstein barycenter problem (i.e. the weighted Frechet mean for the 2-Sinkhorn divergence), formally:

\[\min_\mathbf{X} \quad \sum_{i=1}^N w_i W_{reg}^2(\mathbf{b}, \mathbf{X}, \mathbf{a}_i, \mathbf{X}_i)\]</div>

where :

<li>\(w \in \mathbb{(0, 1)}^{N}\)’s are the barycenter weights and sum to one</li>

<li><cite>measure_weights</cite> denotes the \(\mathbf{a}_i \in \mathbb{R}^{k_i}\): empirical measures weights (on simplex)</li>

<li><cite>measures_locations</cite> denotes the \(\mathbf{X}_i \in \mathbb{R}^{k_i, d}\): empirical measures atoms locations</li>

<li>\(\mathbf{b} \in \mathbb{R}^{k}\) is the desired weights vector of the barycenter</li>

</ul>

This problem is considered in <a class="reference internal" href="#references-free-support-barycenter">[20]</a> (Algorithm 2).

There are two differences with the following codes:

<li>we do not optimize over the weights</li>

<li>we do not do line search for the locations updates, we use i.e. \(\theta = 1\) in

<a class="reference internal" href="#references-free-support-barycenter">[20]</a> (Algorithm 2). This can be seen as a discrete

implementation of the fixed-point algorithm of

<a class="reference internal" href="#references-free-support-barycenter">[43]</a> proposed in the continuous setting.</li>

<li>at each iteration, instead of solving an exact OT problem, we use the Sinkhorn algorithm for calculating the

transport plan in <a class="reference internal" href="#references-free-support-barycenter">[20]</a> (Algorithm 2).</li>

</ul>

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

<li>measures_locations (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#list" title="(in Python v3.14)">list</a> of N (k_i,d) array-like) – The discrete support of a measure supported on \(k_i\) locations of a <cite>d</cite>-dimensional space

(\(k_i\) can be different for each element of the list)</li>

<li>measures_weights (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#list" title="(in Python v3.14)">list</a> of N (k_i,) array-like) – Numpy arrays where each numpy array has \(k_i\) non-negatives values summing to one

representing the weights of each discrete input measure</li>

<li>X_init ((k,d) array-like) – Initialization of the support locations (on <cite>k</cite> atoms) of the barycenter</li>

<li>b ((k,) array-like) – Initialization of the weights of the barycenter (non-negatives, sum to 1)</li>

<li>weights ((N,) array-like) – Initialization of the coefficients of the barycenter (non-negatives, sum to 1)</li>

<li>numInnerItermax (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)">int</a>, optional) – Max number of iterations when calculating the transport plans with Sinkhorn</li>

</ul>

</dd>

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

<dd class="field-even">X – Support locations (on k atoms) of the barycenter

</dd>

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

<dd class="field-odd">(k,d) array-like

</dd>

</dl>

See also

<dt><a class="reference internal" href="#ot.bregman.sinkhorn" title="ot.bregman.sinkhorn"><code class="xref py py-obj docutils literal notranslate">ot.bregman.sinkhorn</code></a></dt><dd>Entropic regularized OT solver

</dd>

<dt><a class="reference internal" href="ot.lp.html#ot.lp.free_support_barycenter" title="ot.lp.free_support_barycenter"><code class="xref py py-obj docutils literal notranslate">ot.lp.free_support_barycenter</code></a></dt><dd>Barycenter solver based on Linear Programming

</dd>

</dl>

</div>

References

[20]

Cuturi, Marco, and Arnaud Doucet. “Fast computation of Wasserstein barycenters.” International Conference on Machine Learning. 2014.

</aside>

[43]

Álvarez-Esteban, Pedro C., et al. “A fixed-point approach to barycenters in Wasserstein space.” Journal of Mathematical Analysis and Applications 441.2 (2016): 744-762.

</aside>

</dd></dl>

ot.bregman.geometricBar(weights, alldistribT)<a class="reference internal" href="../_modules/ot/bregman/_utils.html#geometricBar">[source]</a><a class="headerlink" href="#ot.bregman.geometricBar" title="Link to this definition"></a></dt>

<dd>return the weighted geometric mean of distributions

</dd></dl>

ot.bregman.geometricMean(alldistribT)<a class="reference internal" href="../_modules/ot/bregman/_utils.html#geometricMean">[source]</a><a class="headerlink" href="#ot.bregman.geometricMean" title="Link to this definition"></a></dt>

<dd>return the geometric mean of distributions

</dd></dl>

ot.bregman.greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-09, verbose=False, log=False, warn=True, warmstart=None)<a class="reference internal" href="../_modules/ot/bregman/_sinkhorn.html#greenkhorn">[source]</a><a class="headerlink" href="#ot.bregman.greenkhorn" title="Link to this definition"></a></dt>

<dd>Solve the entropic regularization optimal transport problem and return the OT matrix

The algorithm used is based on the paper <a class="reference internal" href="#references-greenkhorn">[22]</a>

which is a stochastic version of the Sinkhorn-Knopp

algorithm <a class="reference internal" href="#references-greenkhorn">[2]</a>

The function solves the following optimization problem:

\[ \begin{align}\begin{aligned}\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg}\cdot\Omega(\gamma)\\s.t. \ \gamma \mathbf{1} &= \mathbf{a}\\ \gamma^T \mathbf{1} &= \mathbf{b}\\ \gamma &\geq 0\end{aligned}\end{align} \]</div>

where :

<li>\(\mathbf{M}\) is the (<cite>dim_a</cite>, <cite>dim_b</cite>) metric cost matrix</li>

<li>\(\Omega\) is the entropic regularization term

\(\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})\)</li>

<li>\(\mathbf{a}\) and \(\mathbf{b}\) are source and target

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