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<!DOCTYPE html>

<head>

<title>ot.partial — POT Python Optimal Transport 0.9.8.dev documentation</title>

</head>

POT Python Optimal Transport

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<li class="toctree-l2 current"><a class="current reference internal" href="#">ot.partial</a><ul>

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

<li class="toctree-l3"><a class="reference internal" href="#ot.partial.entropic_partial_gromov_wasserstein2"><code class="docutils literal notranslate">entropic_partial_gromov_wasserstein2()</code></a></li>

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

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

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

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

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

<li class="toctree-l3"><a class="reference internal" href="#ot.partial.partial_gromov_wasserstein2"><code class="docutils literal notranslate">partial_gromov_wasserstein2()</code></a></li>

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

<li class="toctree-l3"><a class="reference internal" href="#ot.partial.partial_wasserstein2"><code class="docutils literal notranslate">partial_wasserstein2()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#ot.partial.partial_wasserstein_1d"><code class="docutils literal notranslate">partial_wasserstein_1d()</code></a></li>

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

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

Efficient 1D solver for the partial optimal transport problem.

ot.partial.entropic_partial_gromov_wasserstein(C1, C2, p, q, reg, m=None, G0=None, numItermax=1000, tol=1e-07, log=False, verbose=False)<a class="reference internal" href="../_modules/ot/partial/partial_solvers.html#entropic_partial_gromov_wasserstein">[source]</a><a class="headerlink" href="#ot.partial.entropic_partial_gromov_wasserstein" title="Link to this definition"></a></dt>

<dd>Returns the partial Gromov-Wasserstein transport between

\((\mathbf{C_1}, \mathbf{p})\) and \((\mathbf{C_2}, \mathbf{q})\)

The function solves the following optimization problem:

\[\gamma = \mathop{\arg \min}_{\gamma} \quad \sum_{i,j,k,l}

L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l})\cdot

\gamma_{i,j}\cdot\gamma_{k,l} + \mathrm{reg} \cdot\Omega(\gamma)\]</div>

\[ \begin{align}\begin{aligned}s.t. \ \gamma &\geq 0\\ \gamma \mathbf{1} &\leq \mathbf{a}\\ \gamma^T \mathbf{1} &\leq \mathbf{b}\\ \mathbf{1}^T \gamma^T \mathbf{1} = m

&\leq \min\{\|\mathbf{a}\|_1, \|\mathbf{b}\|_1\}\end{aligned}\end{align} \]</div>

where :

<li>\(\mathbf{C_1}\) is the metric cost matrix in the source space</li>

<li>\(\mathbf{C_2}\) is the metric cost matrix in the target space</li>

<li>\(\mathbf{p}\) and \(\mathbf{q}\) are the sample weights</li>

<li><cite>L</cite>: quadratic loss function</li>

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

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

<li><cite>m</cite> is the amount of mass to be transported</li>

</ul>

The formulation of the GW problem has been proposed in

<a class="reference internal" href="#references-entropic-partial-gromov-wasserstein">[12]</a> and the

partial GW in <a class="reference internal" href="#references-entropic-partial-gromov-wasserstein">[29]</a>

Note

This function will be deprecated in a near future, please use

<cite>ot.gromov.entropic_partial_gromov_wasserstein</cite> instead.

</div>

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

<li>C1 (ndarray, shape (ns, ns)) – Metric cost matrix in the source space</li>

<li>C2 (ndarray, shape (nt, nt)) – Metric cost matrix in the target space</li>

<li>p (ndarray, shape (ns,)) – Distribution in the source space</li>

<li>q (ndarray, shape (nt,)) – Distribution in the target space</li>

<li>reg (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>) – entropic regularization parameter</li>

<li>m (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>, optional) – Amount of mass to be transported (default:

\(\min\{\|\mathbf{p}\|_1, \|\mathbf{q}\|_1\}\))</li>

<li>G0 (ndarray, shape (ns, nt), optional) – Initialization of the transportation matrix</li>

<li>numItermax (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional) – Max 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) – Stop threshold on error (>0)</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) – return log if True</li>

<li>verbose (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#bool" title="(in Python v3.15)">bool</a>, optional) – Print information along iterations</li>

</ul>

</dd>

</dl>

Examples

<div class="doctest highlight-default notranslate"><div class="highlight"><pre>>>> import ot

>>> import scipy as sp

>>> a = np.array([0.25] * 4)

>>> b = np.array([0.25] * 4)

>>> x = np.array([1,2,100,200]).reshape((-1,1))

>>> y = np.array([3,2,98,199]).reshape((-1,1))

>>> C1 = sp.spatial.distance.cdist(x, x)

>>> C2 = sp.spatial.distance.cdist(y, y)

>>> np.round(entropic_partial_gromov_wasserstein(C1, C2, a, b, 50), 2)

array([[0.12, 0.13, 0. , 0. ],

[0.13, 0.12, 0. , 0. ],

[0. , 0. , 0.25, 0. ],

[0. , 0. , 0. , 0.25]])

>>> np.round(entropic_partial_gromov_wasserstein(C1, C2, a, b, 50,0.25), 2)

array([[0.02, 0.03, 0. , 0.03],

[0.03, 0.03, 0. , 0.03],

[0. , 0. , 0.03, 0. ],

[0.02, 0.02, 0. , 0.03]])

</pre></div>

</div>

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

<li>gamma (ndarray, shape (dim_a, dim_b)) – Optimal transportation matrix for the given parameters</li>

<li>log (dict) – log dictionary returned only if <cite>log</cite> is <cite>True</cite></li>

</ul>

</dd>

</dl>

References

[12]

Peyré, Gabriel, Marco Cuturi, and Justin Solomon,

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

International Conference on Machine Learning (ICML). 2016.

</aside>

[29]

Chapel, L., Alaya, M., Gasso, G. (2020). “Partial Optimal

Transport with Applications on Positive-Unlabeled Learning”.

NeurIPS.

</aside>

See also

<dt><a class="reference internal" href="#ot.partial.partial_gromov_wasserstein" title="ot.partial.partial_gromov_wasserstein"><code class="xref py py-obj docutils literal notranslate">ot.partial.partial_gromov_wasserstein</code></a></dt><dd>exact Partial Gromov-Wasserstein

</dd>

</dl>

</div>

</dd></dl>

ot.partial.entropic_partial_gromov_wasserstein2(C1, C2, p, q, reg, m=None, G0=None, numItermax=1000, tol=1e-07, log=False, verbose=False)<a class="reference internal" href="../_modules/ot/partial/partial_solvers.html#entropic_partial_gromov_wasserstein2">[source]</a><a class="headerlink" href="#ot.partial.entropic_partial_gromov_wasserstein2" title="Link to this definition"></a></dt>

<dd>Returns the partial Gromov-Wasserstein discrepancy between

\((\mathbf{C_1}, \mathbf{p})\) and \((\mathbf{C_2}, \mathbf{q})\)

The function solves the following optimization problem:

\[GW = \min_{\gamma} \quad \sum_{i,j,k,l} L(\mathbf{C_1}_{i,k},

\mathbf{C_2}_{j,l})\cdot

\gamma_{i,j}\cdot\gamma_{k,l} + \mathrm{reg} \cdot\Omega(\gamma)\]</div>

\[ \begin{align}\begin{aligned}s.t. \ \gamma &\geq 0\\ \gamma \mathbf{1} &\leq \mathbf{a}\\ \gamma^T \mathbf{1} &\leq \mathbf{b}\\ \mathbf{1}^T \gamma^T \mathbf{1} = m &\leq \min\{\|\mathbf{a}\|_1, \|\mathbf{b}\|_1\}\end{aligned}\end{align} \]</div>

where :

<li>\(\mathbf{C_1}\) is the metric cost matrix in the source space</li>

<li>\(\mathbf{C_2}\) is the metric cost matrix in the target space</li>

<li>\(\mathbf{p}\) and \(\mathbf{q}\) are the sample weights</li>

<li><cite>L</cite> : quadratic loss function</li>

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

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

<li><cite>m</cite> is the amount of mass to be transported</li>

</ul>

The formulation of the GW problem has been proposed in

<a class="reference internal" href="#references-entropic-partial-gromov-wasserstein2">[12]</a> and the

partial GW in <a class="reference internal" href="#references-entropic-partial-gromov-wasserstein2">[29]</a>

Note

This function will be deprecated in a near future, please use

<cite>ot.gromov.entropic_partial_gromov_wasserstein2</cite> instead.

</div>

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

<li>C1 (ndarray, shape (ns, ns)) – Metric cost matrix in the source space</li>

<li>C2 (ndarray, shape (nt, nt)) – Metric cost matrix in the target space</li>

<li>p (ndarray, shape (ns,)) – Distribution in the source space</li>

<li>q (ndarray, shape (nt,)) – Distribution in the target space</li>

\(\min\{\|\mathbf{p}\|_1, \|\mathbf{q}\|_1\}\))</li>

</ul>

</dd>

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

<li>partial_gw_dist (float) – Gromov-Wasserstein distance</li>

<li>log (dict) – log dictionary returned only if <cite>log</cite> is <cite>True</cite></li>

</ul>

</dd>

</dl>

Examples

>>> np.round(entropic_partial_gromov_wasserstein2(C1, C2, a, b,50), 2)

1.87

</pre></div>

</div>

References

[12]

Peyré, Gabriel, Marco Cuturi, and Justin Solomon,

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

International Conference on Machine Learning (ICML). 2016.

</aside>

[29]

Chapel, L., Alaya, M., Gasso, G. (2020). “Partial Optimal

Transport with Applications on Positive-Unlabeled Learning”.

NeurIPS.

</aside>

</dd></dl>

ot.partial.entropic_partial_wasserstein(a, b, M, reg, m=None, method='sinkhorn', numItermax=1000, stopThr=1e-100, verbose=False, log=False)<a class="reference internal" href="../_modules/ot/partial/partial_solvers.html#entropic_partial_wasserstein">[source]</a><a class="headerlink" href="#ot.partial.entropic_partial_wasserstein" title="Link to this definition"></a></dt>

<dd>Solves the partial optimal transport problem

and returns the OT plan

The function considers the following problem:

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

\mathbf{M} \rangle_F + \mathrm{reg} \cdot\Omega(\gamma)\\\begin{split}s.t. \gamma \mathbf{1} &\leq \mathbf{a} \\

\gamma^T \mathbf{1} &\leq \mathbf{b} \\

\gamma &\geq 0 \\

\mathbf{1}^T \gamma^T \mathbf{1} = m

&\leq \min\{\|\mathbf{a}\|_1, \|\mathbf{b}\|_1\} \\\end{split}\end{aligned}\end{align} \]</div>

where :

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

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

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

<li>\(\mathbf{a}\) and \(\mathbf{b}\) are the sample weights</li>

<li><cite>m</cite> is the amount of mass to be transported</li>

</ul>

The formulation of the problem has been proposed in

<a class="reference internal" href="#references-entropic-partial-wasserstein">[3]</a> (prop. 5)

Choosing a solver

By default (<code class="docutils literal notranslate">method='sinkhorn'</code>) the problem is solved with the classical

Sinkhorn-like iterations in the multiplicative (exponential) domain. These

are fast, but the kernel \(K = e^{-\mathbf{M}/\mathrm{reg}}\) underflows

to zero for small <code class="docutils literal notranslate">reg</code> (or large cost values), making the plan collapse

to <code class="docutils literal notranslate">NaN</code>/<code class="docutils literal notranslate">Inf</code> (see Issue #723). When this happens, set

<code class="docutils literal notranslate">method='sinkhorn_log'</code> to switch to the mathematically equivalent

log-domain solver <a class="reference internal" href="#ot.partial.entropic_partial_wasserstein_logscale" title="ot.partial.entropic_partial_wasserstein_logscale"><code class="xref any py py-func docutils literal notranslate">entropic_partial_wasserstein_logscale</code></a>, which relies

on the log-sum-exp trick and stays finite at the price of being slower.

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

<li>a (np.ndarray (dim_a,)) – Unnormalized histogram of dimension <cite>dim_a</cite></li>

<li>b (np.ndarray (dim_b,)) – Unnormalized histograms of dimension <cite>dim_b</cite></li>

<li>M (np.ndarray (dim_a, dim_b)) – cost matrix</li>

<li>method (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)">str</a>, optional) – Solver used, either <code class="docutils literal notranslate">'sinkhorn'</code> (classical multiplicative-domain

solver, default) or <code class="docutils literal notranslate">'sinkhorn_log'</code> (numerically stable log-domain

solver). See the Choosing a solver note above.</li>

<li>stopThr (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>, optional) – Stop threshold on error (>0)</li>

</ul>

</dd>

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

<li>gamma ((dim_a, dim_b) ndarray) – Optimal transportation matrix for the given parameters</li>

<li>log (dict) – log dictionary returned only if <cite>log</cite> is <cite>True</cite></li>

</ul>

</dd>

</dl>

Examples

>>> a = [.1, .2]

>>> b = [.1, .1]

>>> M = [[0., 1.], [2., 3.]]

>>> np.round(entropic_partial_wasserstein(a, b, M, 1, 0.1), 2)

array([[0.06, 0.02],

[0.01, 0. ]])

</pre></div>

</div>

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>

See also

<dt><a class="reference internal" href="#ot.partial.partial_wasserstein" title="ot.partial.partial_wasserstein"><code class="xref py py-obj docutils literal notranslate">ot.partial.partial_wasserstein</code></a></dt><dd>exact Partial Wasserstein

</dd>

<dt><a class="reference internal" href="#ot.partial.entropic_partial_wasserstein_logscale" title="ot.partial.entropic_partial_wasserstein_logscale"><code class="xref py py-obj docutils literal notranslate">ot.partial.entropic_partial_wasserstein_logscale</code></a></dt><dd>stable log-domain solver

</dd>

</dl>

</div>

</dd></dl>

ot.partial.entropic_partial_wasserstein_logscale(a, b, M, reg, m=None, numItermax=1000, stopThr=1e-100, verbose=False, log=False)<a class="reference internal" href="../_modules/ot/partial/partial_solvers.html#entropic_partial_wasserstein_logscale">[source]</a><a class="headerlink" href="#ot.partial.entropic_partial_wasserstein_logscale" title="Link to this definition"></a></dt>

<dd>Solves the partial optimal transport problem

and returns the OT plan

This function solves exactly the same problem as

<a class="reference internal" href="#ot.partial.entropic_partial_wasserstein" title="ot.partial.entropic_partial_wasserstein"><code class="xref any py py-func docutils literal notranslate">entropic_partial_wasserstein</code></a>, but it is in log-scale, i.e., the log-sum-exp trick is used,

so it is more stable but slower.

The input and output format is the same as <a class="reference internal" href="#ot.partial.entropic_partial_wasserstein" title="ot.partial.entropic_partial_wasserstein"><code class="xref any py py-func docutils literal notranslate">entropic_partial_wasserstein</code></a>.

The function considers the following problem:

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

\mathbf{M} \rangle_F + \mathrm{reg} \cdot\Omega(\gamma)\\\begin{split}s.t. \gamma \mathbf{1} &\leq \mathbf{a} \\

\gamma^T \mathbf{1} &\leq \mathbf{b} \\

\gamma &\geq 0 \\

\mathbf{1}^T \gamma^T \mathbf{1} = m

&\leq \min\{\|\mathbf{a}\|_1, \|\mathbf{b}\|_1\} \\\end{split}\end{aligned}\end{align} \]</div>

where :

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

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

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

<li>\(\mathbf{a}\) and \(\mathbf{b}\) are the sample weights</li>

<li><cite>m</cite> is the amount of mass to be transported</li>

</ul>

The formulation of the problem has been proposed in

<a class="reference internal" href="#references-entropic-partial-wasserstein">[3]</a> (prop. 5)

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

<li>a (np.ndarray (dim_a,)) – Unnormalized histogram of dimension <cite>dim_a</cite></li>

<li>b (np.ndarray (dim_b,)) – Unnormalized histograms of dimension <cite>dim_b</cite></li>

<li>M (np.ndarray (dim_a, dim_b)) – cost matrix</li>

</ul>

</dd>

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

<li>gamma ((dim_a, dim_b) ndarray) – Optimal transportation matrix for the given parameters</li>

<li>log (dict) – log dictionary returned only if <cite>log</cite> is <cite>True</cite></li>

</ul>

</dd>

</dl>

Examples

>>> np.round(entropic_partial_wasserstein_logscale(a, b, M, 1, 0.1), 2)

array([[0.06, 0.02],

[0.01, 0. ]])

</pre></div>

</div>

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>

See also

</dd>

<dt><a class="reference internal" href="#ot.partial.entropic_partial_wasserstein" title="ot.partial.entropic_partial_wasserstein"><code class="xref py py-obj docutils literal notranslate">ot.partial.entropic_partial_wasserstein</code></a></dt><dd>classical entropic solver and dispatcher (this solver is also reachable through its <code class="docutils literal notranslate">method='sinkhorn_log'</code> argument)

</dd>

</dl>

</div>

</dd></dl>

ot.partial.gwgrad_partial(C1, C2, T)<a class="reference internal" href="../_modules/ot/partial/partial_solvers.html#gwgrad_partial">[source]</a><a class="headerlink" href="#ot.partial.gwgrad_partial" title="Link to this definition"></a></dt>

<dd>Compute the GW gradient. Note: we can not use the trick in <a class="reference internal" href="#references-gwgrad-partial">[12]</a>

as the marginals may not sum to 1.

Note

This function will be deprecated in a near future, please use

<cite>ot.gromov.gwggrad</cite> instead.

</div>

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

<li>C1 (array of shape (n_p,n_p)) – intra-source (P) cost matrix</li>

<li>C2 (array of shape (n_u,n_u)) – intra-target (U) cost matrix</li>

<li>T (array of shape(n_p+nb_dummies, n_u) (default: None)) – Transport matrix</li>

</ul>

</dd>

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

<dd class="field-even">gradient

</dd>

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

<dd class="field-odd">numpy.array of shape (n_p+nb_dummies, n_u)

</dd>

</dl>

References

[12]

Peyré, Gabriel, Marco Cuturi, and Justin Solomon,

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

International Conference on Machine Learning (ICML). 2016.

</aside>

</dd></dl>

ot.partial.gwloss_partial(C1, C2, T)<a class="reference internal" href="../_modules/ot/partial/partial_solvers.html#gwloss_partial">[source]</a><a class="headerlink" href="#ot.partial.gwloss_partial" title="Link to this definition"></a></dt>

<dd>Compute the GW loss.

Note

This function will be deprecated in a near future, please use

<cite>ot.gromov.gwloss</cite> instead.

</div>

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

<li>C1 (array of shape (n_p,n_p)) – intra-source (P) cost matrix</li>

<li>C2 (array of shape (n_u,n_u)) – intra-target (U) cost matrix</li>

</ul>

</dd>

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

<dd class="field-even">GW loss

</dd>

</dl>

</dd></dl>

ot.partial.partial_gromov_wasserstein(C1, C2, p, q, m=None, nb_dummies=1, G0=None, thres=1, numItermax=1000, tol=1e-07, log=False, verbose=False, **kwargs)<a class="reference internal" href="../_modules/ot/partial/partial_solvers.html#partial_gromov_wasserstein">[source]</a><a class="headerlink" href="#ot.partial.partial_gromov_wasserstein" title="Link to this definition"></a></dt>

<dd>Solves the partial optimal transport problem

and returns the OT plan

The function considers the following problem:

\[\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F\]</div>

\[ \begin{align}\begin{aligned}s.t. \ \gamma \mathbf{1} &\leq \mathbf{a}\\ \gamma^T \mathbf{1} &\leq \mathbf{b}\\ \gamma &\geq 0\\ \mathbf{1}^T \gamma^T \mathbf{1} = m &\leq \min\{\|\mathbf{a}\|_1, \|\mathbf{b}\|_1\}\end{aligned}\end{align} \]</div>

where :

<li>\(\mathbf{M}\) is the 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 the sample weights</li>

<li><cite>m</cite> is the amount of mass to be transported</li>

</ul>

The formulation of the problem has been proposed in

Note

This function will be deprecated in a near future, please use

<cite>ot.gromov.partial_gromov_wasserstein</cite> instead.

</div>

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

<li>C1 (ndarray, shape (ns, ns)) – Metric cost matrix in the source space</li>

<li>C2 (ndarray, shape (nt, nt)) – Metric costfr matrix in the target space</li>

<li>p (ndarray, shape (ns,)) – Distribution in the source space</li>

<li>q (ndarray, shape (nt,)) – Distribution in the target space</li>

(default: \(\min\{\|\mathbf{p}\|_1, \|\mathbf{q}\|_1\}\))</li>

<li>nb_dummies (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional) – Number of dummy points to add (avoid instabilities in the EMD solver)</li>

<li>thres (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>, optional) – quantile of the gradient matrix to populate the cost matrix when 0

(default: 1)</li>

<li>**kwargs (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#dict" title="(in Python v3.15)">dict</a>) – parameters can be directly passed to the emd solver</li>

</ul>

</dd>

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

<li>gamma ((dim_a, dim_b) ndarray) – Optimal transportation matrix for the given parameters</li>

<li>log (dict) – log dictionary returned only if <cite>log</cite> is <cite>True</cite></li>

</ul>

</dd>

</dl>

Examples

>>> np.round(partial_gromov_wasserstein(C1, C2, a, b),2)

array([[0. , 0.25, 0. , 0. ],

[0.25, 0. , 0. , 0. ],

[0. , 0. , 0.25, 0. ],

[0. , 0. , 0. , 0.25]])

>>> np.round(partial_gromov_wasserstein(C1, C2, a, b, m=0.25),2)

array([[0. , 0. , 0. , 0. ],

[0. , 0. , 0. , 0. ],

[0. , 0. , 0.25, 0. ],

[0. , 0. , 0. , 0. ]])

</pre></div>

</div>

References

[29]

Chapel, L., Alaya, M., Gasso, G. (2020). “Partial Optimal

Transport with Applications on Positive-Unlabeled Learning”.

NeurIPS.

</aside>

</dd></dl>

ot.partial.partial_gromov_wasserstein2(C1, C2, p, q, m=None, nb_dummies=1, G0=None, thres=1, numItermax=1000, tol=1e-07, log=False, verbose=False, **kwargs)<a class="reference internal" href="../_modules/ot/partial/partial_solvers.html#partial_gromov_wasserstein2">[source]</a><a class="headerlink" href="#ot.partial.partial_gromov_wasserstein2" title="Link to this definition"></a></dt>

<dd>Solves the partial optimal transport problem

and returns the partial Gromov-Wasserstein discrepancy

The function considers the following problem:

\[GW = \min_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F\]</div>

\[ \begin{align}\begin{aligned}s.t. \ \gamma \mathbf{1} &\leq \mathbf{a}\\ \gamma^T \mathbf{1} &\leq \mathbf{b}\\ \gamma &\geq 0\\ \mathbf{1}^T \gamma^T \mathbf{1} = m

&\leq \min\{\|\mathbf{a}\|_1, \|\mathbf{b}\|_1\}\end{aligned}\end{align} \]</div>

where :

<li>\(\mathbf{M}\) is the 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 the sample weights</li>

<li><cite>m</cite> is the amount of mass to be transported</li>

</ul>

The formulation of the problem has been proposed in

Note

This function will be deprecated in a near future, please use

<cite>ot.gromov.partial_gromov_wasserstein2</cite> instead.

</div>

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

<li>C1 (ndarray, shape (ns, ns)) – Metric cost matrix in the source space</li>

<li>C2 (ndarray, shape (nt, nt)) – Metric cost matrix in the target space</li>

<li>p (ndarray, shape (ns,)) – Distribution in the source space</li>

<li>q (ndarray, shape (nt,)) – Distribution in the target space</li>

(default: \(\min\{\|\mathbf{p}\|_1, \|\mathbf{q}\|_1\}\))</li>

(default: 1)</li>

</ul>

</dd>

</dl>

Warning

When dealing with a large number of points, the EMD solver may face

some instabilities, especially when the mass associated to the dummy

point is large. To avoid them, increase the number of dummy points

(allows a smoother repartition of the mass over the points).

</div>

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

<li>partial_gw_dist (float) – partial GW discrepancy</li>

<li>log (dict) – log dictionary returned only if <cite>log</cite> is <cite>True</cite></li>

</ul>

</dd>

</dl>

Examples

>>> np.round(partial_gromov_wasserstein2(C1, C2, a, b),2)

1.69

>>> np.round(partial_gromov_wasserstein2(C1, C2, a, b, m=0.25),2)

0.0

</pre></div>

</div>

References

[29]

Chapel, L., Alaya, M., Gasso, G. (2020). “Partial Optimal

Transport with Applications on Positive-Unlabeled Learning”.

NeurIPS.

</aside>

</dd></dl>

ot.partial.partial_wasserstein(a, b, M, m=None, nb_dummies=1, log=False, **kwargs)<a class="reference internal" href="../_modules/ot/partial/partial_solvers.html#partial_wasserstein">[source]</a><a class="headerlink" href="#ot.partial.partial_wasserstein" title="Link to this definition"></a></dt>

<dd>Solves the partial optimal transport problem for the quadratic cost

and returns the OT plan

The function considers the following problem:

\[\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F\]</div>

where :

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

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

<li><cite>m</cite> is the amount of mass to be transported</li>

</ul>

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

<li>a (np.ndarray (dim_a,)) – Unnormalized histogram of dimension <cite>dim_a</cite></li>

<li>b (np.ndarray (dim_b,)) – Unnormalized histograms of dimension <cite>dim_b</cite></li>

<li>M (np.ndarray (dim_a, dim_b)) – cost matrix for the quadratic cost</li>

<li>nb_dummies (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional, default:1) – number of reservoir points to be added (to avoid numerical

instabilities, increase its value if an error is raised)</li>

</ul>

</dd>

</dl>

Warning

When dealing with a large number of points, the EMD solver may face

some instabilities, especially when the mass associated to the dummy

point is large. To avoid them, increase the number of dummy points

(allows a smoother repartition of the mass over the points).

</div>

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

<li>gamma ((dim_a, dim_b) ndarray) – Optimal transportation matrix for the given parameters</li>

<li>log (dict) – log dictionary returned only if <cite>log</cite> is <cite>True</cite></li>

</ul>

</dd>

</dl>

Examples

>>> np.round(partial_wasserstein(a,b,M), 2)

array([[0.1, 0. ],

[0. , 0.1]])

>>> np.round(partial_wasserstein(a,b,M,m=0.1), 2)

array([[0.1, 0. ],

[0. , 0. ]])

</pre></div>

</div>

References

[28]

Caffarelli, L. A., & McCann, R. J. (2010) Free boundaries in

optimal transport and Monge-Ampere obstacle problems. Annals of

mathematics, 673-730.

</aside>

[29]

Chapel, L., Alaya, M., Gasso, G. (2020). “Partial Optimal

Transport with Applications on Positive-Unlabeled Learning”.

NeurIPS.

</aside>

See also

<dt><a class="reference internal" href="#ot.partial.partial_wasserstein_lagrange" title="ot.partial.partial_wasserstein_lagrange"><code class="xref py py-obj docutils literal notranslate">ot.partial.partial_wasserstein_lagrange</code></a></dt><dd>Partial Wasserstein with regularization on the marginals

</dd>

<dt><a class="reference internal" href="#ot.partial.entropic_partial_wasserstein" title="ot.partial.entropic_partial_wasserstein"><code class="xref py py-obj docutils literal notranslate">ot.partial.entropic_partial_wasserstein</code></a></dt><dd>Partial Wasserstein with a entropic regularization parameter

</dd>

</dl>

</div>

</dd></dl>

ot.partial.partial_wasserstein2(a, b, M, m=None, nb_dummies=1, log=False, **kwargs)<a class="reference internal" href="../_modules/ot/partial/partial_solvers.html#partial_wasserstein2">[source]</a><a class="headerlink" href="#ot.partial.partial_wasserstein2" title="Link to this definition"></a></dt>

<dd>Solves the partial optimal transport problem for the quadratic cost

and returns the partial GW discrepancy

The function considers the following problem:

\[\gamma = \min_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F\]</div>

where :

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

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

<li><cite>m</cite> is the amount of mass to be transported</li>

</ul>

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

<li>a (np.ndarray (dim_a,)) – Unnormalized histogram of dimension <cite>dim_a</cite></li>

<li>b (np.ndarray (dim_b,)) – Unnormalized histograms of dimension <cite>dim_b</cite></li>

<li>M (np.ndarray (dim_a, dim_b)) – cost matrix for the quadratic cost</li>

<li>nb_dummies (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional, default:1) – number of reservoir points to be added (to avoid numerical

instabilities, increase its value if an error is raised)</li>

</ul>

</dd>

</dl>

Warning

When dealing with a large number of points, the EMD solver may face

some instabilities, especially when the mass associated to the dummy

point is large. To avoid them, increase the number of dummy points

(allows a smoother repartition of the mass over the points).

</div>

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

<li>GW (float) – partial GW discrepancy</li>

<li>log (dict) – log dictionary returned only if <cite>log</cite> is <cite>True</cite></li>

</ul>

</dd>

</dl>

Examples

>>> a=[.1, .2]

>>> b=[.1, .1]

>>> M=[[0., 1.], [2., 3.]]

>>> np.round(partial_wasserstein2(a, b, M), 1)

0.3

>>> np.round(partial_wasserstein2(a,b,M,m=0.1), 1)

0.0

</pre></div>

</div>

References

[28]

Caffarelli, L. A., & McCann, R. J. (2010) Free boundaries in

optimal transport and Monge-Ampere obstacle problems. Annals of

mathematics, 673-730.

</aside>

[29]

Chapel, L., Alaya, M., Gasso, G. (2020). “Partial Optimal

Transport with Applications on Positive-Unlabeled Learning”.

NeurIPS.

</aside>

</dd></dl>

ot.partial.partial_wasserstein_1d(x_a, x_b, n_transported_samples=None, p=1)<a class="reference internal" href="../_modules/ot/partial/partial_solvers.html#partial_wasserstein_1d">[source]</a><a class="headerlink" href="#ot.partial.partial_wasserstein_1d" title="Link to this definition"></a></dt>

<dd>Solves the partial Wasserstein distance problem between 1d measures and returns

all required information to reconstruct the solutions to all partial problems for any

amount of transported samples up to <cite>n_transported_samples</cite>.

The function considers the following problem:

\[\gamma = \mathop{\arg \min}_\gamma \quad \sum_{ij} \gamma_{ij} \|x_a[i] - x_b[j]\|^p_p\]</div>

\[ \begin{align}\begin{aligned}s.t. \ \gamma \mathbf{1} &\leq \mathbf{1}\\ \gamma^T \mathbf{1} &\leq \mathbf{1}\\ \gamma &\geq 0\\ \mathbf{1}^T \gamma^T \mathbf{1} = n_\text{transported samples}\end{aligned}\end{align} \]</div>

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

<li>x_a (ndarray of float64, shape (n,) or (n, 1)) – Source dirac locations (on the real line)</li>

<li>x_b (ndarray of float64, shape (m,) or (m, 1)) – Target dirac locations (on the real line)</li>

<li>n_transported_samples (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional (default: min(n, m))) – number of samples to be transported</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) – power of the metric (default: 1)</li>

</ul>

</dd>

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

<li>indices_x (np.ndarray of shape (min(n, m, n_transported_samples), )) – Indices of elements from the x distribution to be included in the partial solutions

Order of appearance in this array indicates order of inclusion in the solution</li>

<li>indices_y (np.ndarray of shape (min(n, m, n_transported_samples), )) – Indices of elements from the y distribution to be included in the partial solutions

Order of appearance in this array indicates order of inclusion in the solution</li>

<li>list_marginal_costs (list of length min(n, m, n_transported_samples)) – List of marginal costs associated to the intermediate partial problems

<cite>np.cumsum(list_marginal_costs)</cite> gives the corresponding total costs for intermediate partial problems</li>

</ul>

</dd>

</dl>

Examples

<div class="doctest highlight-default notranslate"><div class="highlight"><pre>>>> x = np.array([5., -2., 4.])

>>> ind_x, ind_y, marginal_costs = partial_wasserstein_1d(x, y, n_transported_samples=2)

>>> ind_x

array([1, 2])

>>> ind_y

array([0, 2])

>>> np.sum(marginal_costs)

2.0

>>> np.sum(np.abs(np.sort(x[ind_x]) - np.sort(y[ind_y])))

2.0

</pre></div>

</div>

References

[76]

Chapel, L., Tavenard, R. (2025).

“One for all and all for one:

Efficient computation of partial Wasserstein distances on the line”.

ICLR.

</aside>

See also

</dd>

</dl>

</div>

</dd></dl>

ot.partial.partial_wasserstein_lagrange(a, b, M, reg_m=None, nb_dummies=1, log=False, **kwargs)<a class="reference internal" href="../_modules/ot/partial/partial_solvers.html#partial_wasserstein_lagrange">[source]</a><a class="headerlink" href="#ot.partial.partial_wasserstein_lagrange" title="Link to this definition"></a></dt>

<dd>Solves the partial optimal transport problem for the quadratic cost

and returns the OT plan

The function considers the following problem:

\[\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, (\mathbf{M} - \lambda) \rangle_F\]</div>

\[ \begin{align}\begin{aligned}s.t. \ \gamma \mathbf{1} &\leq \mathbf{a}\\ \gamma^T \mathbf{1} &\leq \mathbf{b}\\ \gamma &\geq 0\\ \mathbf{1}^T \gamma^T \mathbf{1} = m &

\leq \min\{\|\mathbf{a}\|_1, \|\mathbf{b}\|_1\}\end{aligned}\end{align} \]</div>

or equivalently (see Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X.

(2018). An interpolating distance between optimal transport and Fisher–Rao

metrics. Foundations of Computational Mathematics, 18(1), 1-44.)

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

\sqrt{\frac{\lambda}{2} (\|\gamma \mathbf{1} - \mathbf{a}\|_1 + \|\gamma^T \mathbf{1} - \mathbf{b}\|_1)}\\s.t. \ \gamma \geq 0\end{aligned}\end{align} \]</div>

where :

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

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

<li>\(\lambda\) is the lagrangian cost. Tuning its value allows attaining

a given mass to be transported <cite>m</cite></li>

</ul>

The formulation of the problem has been proposed in

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

<li>a (np.ndarray (dim_a,)) – Unnormalized histogram of dimension <cite>dim_a</cite></li>

<li>b (np.ndarray (dim_b,)) – Unnormalized histograms of dimension <cite>dim_b</cite></li>

<li>M (np.ndarray (dim_a, dim_b)) – cost matrix for the quadratic cost</li>

<li>reg_m (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>, optional) – Lagrangian cost</li>

<li>nb_dummies (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional, default:1) – number of reservoir points to be added (to avoid numerical

instabilities, increase its value if an error is raised)</li>

</ul>

</dd>

</dl>

Warning

When dealing with a large number of points, the EMD solver may face

some instabilities, especially when the mass associated to the dummy

point is large. To avoid them, increase the number of dummy points

(allows a smoother repartition of the mass over the points).

</div>

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

<li>gamma ((dim_a, dim_b) ndarray) – Optimal transportation matrix for the given parameters</li>

<li>log (dict) – log dictionary returned only if <cite>log</cite> is <cite>True</cite></li>

</ul>

</dd>

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