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<!DOCTYPE html>

<head>

<title>API and modules — POT Python Optimal Transport 0.7.0 documentation</title>

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</head>

<a href="index.html" class="icon icon-home" alt="Documentation Home"> POT Python Optimal Transport

</a>

0.7.0

</div>

</form>

</div>

<li class="toctree-l1"><a class="reference internal" href="index.html">POT: Python Optimal Transport</a></li>

<li class="toctree-l1"><a class="reference internal" href="quickstart.html">Quick start guide</a></li>

<li class="toctree-l1 current"><a class="current reference internal" href="#">API and modules</a><ul>

<li class="toctree-l2"><a class="reference internal" href="gen_modules/ot.bregman.html">ot.bregman</a></li>

<li class="toctree-l2"><a class="reference internal" href="gen_modules/ot.smooth.html">ot.smooth</a></li>

<li class="toctree-l2"><a class="reference internal" href="gen_modules/ot.gromov.html">ot.gromov</a></li>

<li class="toctree-l2"><a class="reference internal" href="gen_modules/ot.optim.html">ot.optim</a></li>

<li class="toctree-l2"><a class="reference internal" href="gen_modules/ot.utils.html">ot.utils</a></li>

<li class="toctree-l2"><a class="reference internal" href="gen_modules/ot.datasets.html">ot.datasets</a></li>

<li class="toctree-l2"><a class="reference internal" href="gen_modules/ot.stochastic.html">ot.stochastic</a></li>

<li class="toctree-l2"><a class="reference internal" href="gen_modules/ot.unbalanced.html">ot.unbalanced</a></li>

<li class="toctree-l2"><a class="reference internal" href="gen_modules/ot.partial.html">ot.partial</a></li>

</ul>

</li>

<li class="toctree-l1"><a class="reference internal" href="auto_examples/index.html">Examples gallery</a></li>

<li class="toctree-l1"><a class="reference internal" href="releases.html">Releases</a></li>

</ul>

</div>

</nav>

<a href="index.html">POT Python Optimal Transport</a>

</nav>

<li>API and modules</li>

<a href="_sources/all.rst.txt" rel="nofollow"> View page source</a>

</li>

</ul>

<hr/>

</div>

<h1>API and modules<a class="headerlink" href="#api-and-modules" title="Permalink to this headline">¶</a></h1>

<a class="reference internal" href="#module-ot" title="ot"><code class="xref py py-mod docutils literal notranslate">ot</code></a>:

</colgroup>

<tbody>

<td>Solvers for the original linear program OT problem</td>

</tr>

<tr class="row-even"><td><a class="reference internal" href="gen_modules/ot.bregman.html#module-ot.bregman" title="ot.bregman"><code class="xref py py-obj docutils literal notranslate">bregman</code></a></td>

<td>Bregman projections solvers for entropic regularized OT</td>

</tr>

<tr class="row-odd"><td><a class="reference internal" href="gen_modules/ot.smooth.html#module-ot.smooth" title="ot.smooth"><code class="xref py py-obj docutils literal notranslate">smooth</code></a></td>

<td>Smooth and Sparse Optimal Transport solvers (KL an L2 reg.)</td>

</tr>

<tr class="row-even"><td><a class="reference internal" href="gen_modules/ot.gromov.html#module-ot.gromov" title="ot.gromov"><code class="xref py py-obj docutils literal notranslate">gromov</code></a></td>

<td>Gromov-Wasserstein and Fused-Gromov-Wasserstein solvers</td>

</tr>

<tr class="row-odd"><td><a class="reference internal" href="gen_modules/ot.optim.html#module-ot.optim" title="ot.optim"><code class="xref py py-obj docutils literal notranslate">optim</code></a></td>

<td>Generic solvers for regularized OT</td>

</tr>

<td>Domain adaptation with optimal transport</td>

</tr>

<td>GPU implementation for several OT solvers and utility functions.</td>

</tr>

<td>Dimension reduction with OT</td>

</tr>

<tr class="row-odd"><td><a class="reference internal" href="gen_modules/ot.utils.html#module-ot.utils" title="ot.utils"><code class="xref py py-obj docutils literal notranslate">utils</code></a></td>

<td>Various useful functions</td>

</tr>

<tr class="row-even"><td><a class="reference internal" href="gen_modules/ot.datasets.html#module-ot.datasets" title="ot.datasets"><code class="xref py py-obj docutils literal notranslate">datasets</code></a></td>

<td>Simple example datasets</td>

</tr>

<td>Functions for plotting OT matrices</td>

</tr>

<tr class="row-even"><td><a class="reference internal" href="gen_modules/ot.stochastic.html#module-ot.stochastic" title="ot.stochastic"><code class="xref py py-obj docutils literal notranslate">stochastic</code></a></td>

<td>Stochastic solvers for regularized OT.</td>

</tr>

<tr class="row-odd"><td><a class="reference internal" href="gen_modules/ot.unbalanced.html#module-ot.unbalanced" title="ot.unbalanced"><code class="xref py py-obj docutils literal notranslate">unbalanced</code></a></td>

<td>Regularized Unbalanced OT solvers</td>

</tr>

<tr class="row-even"><td><a class="reference internal" href="gen_modules/ot.partial.html#module-ot.partial" title="ot.partial"><code class="xref py py-obj docutils literal notranslate">partial</code></a></td>

<td>Partial OT solvers</td>

</tr>

</tbody>

</table>

</colgroup>

<tbody>

</tbody>

</table>

Warning

The list of automatically imported sub-modules is as follows:

<a class="reference internal" href="gen_modules/ot.lp.html#module-ot.lp" title="ot.lp"><code class="xref py py-mod docutils literal notranslate">ot.lp</code></a>, <a class="reference internal" href="gen_modules/ot.bregman.html#module-ot.bregman" title="ot.bregman"><code class="xref py py-mod docutils literal notranslate">ot.bregman</code></a>, <a class="reference internal" href="gen_modules/ot.optim.html#module-ot.optim" title="ot.optim"><code class="xref py py-mod docutils literal notranslate">ot.optim</code></a>

<a class="reference internal" href="gen_modules/ot.utils.html#module-ot.utils" title="ot.utils"><code class="xref py py-mod docutils literal notranslate">ot.utils</code></a>, <a class="reference internal" href="gen_modules/ot.datasets.html#module-ot.datasets" title="ot.datasets"><code class="xref py py-mod docutils literal notranslate">ot.datasets</code></a>,

<a class="reference internal" href="gen_modules/ot.gromov.html#module-ot.gromov" title="ot.gromov"><code class="xref py py-mod docutils literal notranslate">ot.gromov</code></a>, <a class="reference internal" href="gen_modules/ot.smooth.html#module-ot.smooth" title="ot.smooth"><code class="xref py py-mod docutils literal notranslate">ot.smooth</code></a>

<a class="reference internal" href="gen_modules/ot.stochastic.html#module-ot.stochastic" title="ot.stochastic"><code class="xref py py-mod docutils literal notranslate">ot.stochastic</code></a>

The following sub-modules are not imported due to additional dependencies:

<li><a class="reference internal" href="gen_modules/ot.dr.html#module-ot.dr" title="ot.dr"><code class="xref any py py-mod docutils literal notranslate">ot.dr</code></a> : depends on <code class="code docutils literal notranslate">pymanopt</code> and <code class="code docutils literal notranslate">autograd</code>.</li>

<li><a class="reference internal" href="gen_modules/ot.gpu.html#module-ot.gpu" title="ot.gpu"><code class="xref any py py-mod docutils literal notranslate">ot.gpu</code></a> : depends on <code class="code docutils literal notranslate">cupy</code> and a CUDA GPU.</li>

<li><a class="reference internal" href="gen_modules/ot.plot.html#module-ot.plot" title="ot.plot"><code class="xref any py py-mod docutils literal notranslate">ot.plot</code></a> : depends on <code class="code docutils literal notranslate">matplotlib</code></li>

</ul>

</div>

<code class="sig-prename descclassname">ot.</code><code class="sig-name descname">barycenter</code>(A, M, reg, weights=None, method='sinkhorn', numItermax=10000, stopThr=0.0001, verbose=False, log=False, **kwargs)<a class="reference internal" href="_modules/ot/bregman.html#barycenter">[source]</a><a class="headerlink" href="#ot.barycenter" title="Permalink to this definition">¶</a></dt>

<dd>Compute the entropic regularized wasserstein barycenter of distributions A

<div>The function solves the following optimization problem:

</div></blockquote>

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

where :

<li>\(W_{reg}(\cdot,\cdot)\) is the entropic regularized Wasserstein distance (see ot.bregman.sinkhorn)</li>

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

<li>reg 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 href="#id38">[3]_</a>

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

<li>A (ndarray, shape (dim, n_hists)) – n_hists training distributions a_i of size dim</li>

<li>M (ndarray, 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 v3.8)">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 v3.8)">str</a> (optional)) – method used for the solver either ‘sinkhorn’ or ‘sinkhorn_stabilized’</li>

<li>weights (ndarray, shape (n_hists,)) – Weights of each histogram a_i on the simplex (barycentric coodinates)</li>

<li>numItermax (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.8)">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 v3.8)">float</a>, optional) – Stop threshol on error (>0)</li>

<li>verbose (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)">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 v3.8)">bool</a>, optional) – record log if True</li>

</ul>

</dd>

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

<li>a ((dim,) ndarray) – Wasserstein barycenter</li>

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

</ul>

</dd>

</dl>

References

<dd>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.

</dd>

</dl>

</dd></dl>

<code class="sig-prename descclassname">ot.</code><code class="sig-name descname">barycenter_unbalanced</code>(A, M, reg, reg_m, method='sinkhorn', weights=None, numItermax=1000, stopThr=1e-06, verbose=False, log=False, **kwargs)<a class="reference internal" href="_modules/ot/unbalanced.html#barycenter_unbalanced">[source]</a><a class="headerlink" href="#ot.barycenter_unbalanced" title="Permalink to this definition">¶</a></dt>

<dd>Compute the entropic unbalanced wasserstein barycenter of A.

<div>The function solves the following optimization problem with a

</div></blockquote>

\[\mathbf{a} = arg\min_\mathbf{a} \sum_i Wu_{reg}(\mathbf{a},\mathbf{a}_i)\]</div>

where :

<li>\(Wu_{reg}(\cdot,\cdot)\) is the unbalanced entropic regularized</li>

</ul>

Wasserstein distance (see ot.unbalanced.sinkhorn_unbalanced)

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

\(\mathbf{A}\)

- reg and \(\mathbf{M}\) are respectively the regularization term and

the cost matrix for OT

- reg_mis the marginal relaxation hyperparameter

The algorithm used for solving the problem is the generalized

Sinkhorn-Knopp matrix scaling algorithm as proposed in <a href="#id39">[10]_</a>

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

<li>A (np.ndarray (dim, n_hists)) – <cite>n_hists</cite> training distributions a_i of dimension dim</li>

<li>M (np.ndarray (dim, dim)) – ground metric matrix for OT.</li>

<li>reg_m (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.8)">float</a>) – Marginal relaxation term > 0</li>

<li>weights (np.ndarray (n_hists,) optional) – Weight of each distribution (barycentric coodinates)

If None, uniform weights are used.</li>

</ul>

</dd>

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

<li>a ((dim,) ndarray) – Unbalanced Wasserstein barycenter</li>

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

</ul>

</dd>

</dl>

References

<dd>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.

</dd>

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

Scaling algorithms for unbalanced transport problems. arXiv preprin

arXiv:1607.05816.

</dd>

</dl>

</dd></dl>

<code class="sig-prename descclassname">ot.</code><code class="sig-name descname">dist</code>(x1, x2=None, metric='sqeuclidean')<a class="reference internal" href="_modules/ot/utils.html#dist">[source]</a><a class="headerlink" href="#ot.dist" title="Permalink to this definition">¶</a></dt>

<dd>Compute distance between samples in x1 and x2 using function scipy.spatial.distance.cdist

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

<li>x1 (ndarray, shape (n1,d)) – matrix with n1 samples of size d</li>

<li>x2 (array, shape (n2,d), optional) – matrix with n2 samples of size d (if None then x2=x1)</li>

<li>metric (str | callable, optional) – Name of the metric to be computed (full list in the doc of scipy), If a string,

the distance function can be ‘braycurtis’, ‘canberra’, ‘chebyshev’, ‘cityblock’,

‘correlation’, ‘cosine’, ‘dice’, ‘euclidean’, ‘hamming’, ‘jaccard’, ‘kulsinski’,

‘mahalanobis’, ‘matching’, ‘minkowski’, ‘rogerstanimoto’, ‘russellrao’, ‘seuclidean’,

‘sokalmichener’, ‘sokalsneath’, ‘sqeuclidean’, ‘wminkowski’, ‘yule’.</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">np.array (n1,n2)

</dd>

</dl>

</dd></dl>

<code class="sig-prename descclassname">ot.</code><code class="sig-name descname">emd</code>(a, b, M, numItermax=100000, log=False, center_dual=True)<a class="reference internal" href="_modules/ot/lp.html#emd">[source]</a><a class="headerlink" href="#ot.emd" title="Permalink to this definition">¶</a></dt>

<dd>Solves the Earth Movers distance problem and returns the OT matrix

\[ \begin{align}\begin{aligned}\gamma = arg\min_\gamma <\gamma,M>_F\\s.t. \gamma 1 = a\\ \gamma^T 1= b\\ \gamma\geq 0\end{aligned}\end{align} \]</div>

where :

<li>M is the metric cost matrix</li>

<li>a and b are the sample weights</li>

</ul>

Warning

Note that the M matrix needs to be a C-order numpy.array in float64

format.

</div>

Uses the algorithm proposed in <a href="#id40">[1]_</a>

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

<li>a ((ns,) <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v1.19)">numpy.ndarray</a>, float64) – Source histogram (uniform weight if empty list)</li>

<li>b ((nt,) <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v1.19)">numpy.ndarray</a>, float64) – Target histogram (uniform weight if empty list)</li>

<li>M ((ns,nt) <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v1.19)">numpy.ndarray</a>, float64) – Loss matrix (c-order array with type float64)</li>

algorithm if it has not converged.</li>

variables. Otherwise returns only the optimal transportation matrix.</li>

<li>center_dual (boolean, optional (default=True)) – If True, centers the dual potential using function

center_ot_dual.</li>

</ul>

</dd>

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

<li>gamma ((ns x nt) numpy.ndarray) – Optimal transportation matrix for the given parameters</li>

<li>log (dict) – If input log is true, a dictionary containing the cost and dual

variables and exit status</li>

</ul>

</dd>

</dl>

Examples

Simple example with obvious solution. The function emd accepts lists and

perform automatic conversion to numpy arrays

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

>>> a=[.5,.5]

>>> b=[.5,.5]

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

>>> ot.emd(a,b,M)

array([[0.5, 0. ],

[0. , 0.5]])

</pre></div>

</div>

References

<dd>Bonneel, N., Van De Panne, M., Paris, S., & Heidrich, W.

(2011, December). Displacement interpolation using Lagrangian mass

transport. In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p.

158). ACM.

</dd>

</dl>

See also

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

</dd>

<dt><a class="reference internal" href="gen_modules/ot.optim.html#ot.optim.cg" title="ot.optim.cg"><code class="xref py py-func docutils literal notranslate">ot.optim.cg()</code></a></dt><dd>General regularized OT

</dd>

</dl>

</div>

</dd></dl>

<code class="sig-prename descclassname">ot.</code><code class="sig-name descname">emd2</code>(a, b, M, processes=36, numItermax=100000, log=False, return_matrix=False, center_dual=True)<a class="reference internal" href="_modules/ot/lp.html#emd2">[source]</a><a class="headerlink" href="#ot.emd2" title="Permalink to this definition">¶</a></dt>

<dd>Solves the Earth Movers distance problem and returns the loss

\[ \begin{align}\begin{aligned}\min_\gamma <\gamma,M>_F\\s.t. \gamma 1 = a\\ \gamma^T 1= b\\ \gamma\geq 0\end{aligned}\end{align} \]</div>

where :

<li>M is the metric cost matrix</li>

<li>a and b are the sample weights</li>

</ul>

Warning

Note that the M matrix needs to be a C-order numpy.array in float64

format.

</div>

Uses the algorithm proposed in <a href="#id41">[1]_</a>

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

<li>processes (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.8)">int</a>, optional (default=nb cpu)) – Nb of processes used for multiple emd computation (not used on windows)</li>

algorithm if it has not converged.</li>

<li>log (boolean, optional (default=False)) – If True, returns a dictionary containing the cost and dual

variables. Otherwise returns only the optimal transportation cost.</li>

<li>return_matrix (boolean, optional (default=False)) – If True, returns the optimal transportation matrix in the log.</li>

<li>center_dual (boolean, optional (default=True)) – If True, centers the dual potential using function

center_ot_dual.</li>

</ul>

</dd>

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

<li>gamma ((ns x nt) ndarray) – Optimal transportation matrix for the given parameters</li>

<li>log (dictnp) – If input log is true, a dictionary containing the cost and dual

variables and exit status</li>

</ul>

</dd>

</dl>

Examples

Simple example with obvious solution. The function emd accepts lists and

perform automatic conversion to numpy arrays

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

>>> ot.emd2(a,b,M)

0.0

</pre></div>

</div>

References

<dd>Bonneel, N., Van De Panne, M., Paris, S., & Heidrich, W.

(2011, December). Displacement interpolation using Lagrangian mass

transport. In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p.

158). ACM.

</dd>

</dl>

See also

</dd>

</dl>

</div>

</dd></dl>

<code class="sig-prename descclassname">ot.</code><code class="sig-name descname">emd2_1d</code>(x_a, x_b, a=None, b=None, metric='sqeuclidean', p=1.0, dense=True, log=False)<a class="reference internal" href="_modules/ot/lp.html#emd2_1d">[source]</a><a class="headerlink" href="#ot.emd2_1d" title="Permalink to this definition">¶</a></dt>

<dd>Solves the Earth Movers distance problem between 1d measures and returns

the loss

\[ \begin{align}\begin{aligned}\gamma = arg\min_\gamma \sum_i \sum_j \gamma_{ij} d(x_a[i], x_b[j])\\s.t. \gamma 1 = a,

\gamma^T 1= b,

\gamma\geq 0\end{aligned}\end{align} \]</div>

where :

<li>d is the metric</li>

<li>x_a and x_b are the samples</li>

<li>a and b are the sample weights</li>

</ul>

When ‘minkowski’ is used as a metric, \(d(x, y) = |x - y|^p\).

Uses the algorithm detailed in <a href="#id42">[1]_</a>

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

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

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

<li>a ((ns,) ndarray, float64, optional) – Source histogram (default is uniform weight)</li>

<li>b ((nt,) ndarray, float64, optional) – Target histogram (default is uniform weight)</li>

<li>metric (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.8)">str</a>, optional (default='sqeuclidean')) – Metric to be used. Only strings listed in <a class="reference internal" href="#ot.dist" title="ot.dist"><code class="xref py py-func docutils literal notranslate">ot.dist()</code></a> are accepted.

Due to implementation details, this function runs faster when

<cite>‘sqeuclidean’</cite>, <cite>‘minkowski’</cite>, <cite>‘cityblock’</cite>, or <cite>‘euclidean’</cite> metrics

are used.</li>

<li>p (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.8)">float</a>, optional (default=1.0)) – The p-norm to apply for if metric=’minkowski’</li>

<li>dense (boolean, optional (default=True)) – If True, returns math:<cite>gamma</cite> as a dense ndarray of shape (ns, nt).

Otherwise returns a sparse representation using scipy’s <cite>coo_matrix</cite>

format. Only used if log is set to True. Due to implementation details,

this function runs faster when dense is set to False.</li>

<li>log (boolean, optional (default=False)) – If True, returns a dictionary containing the transportation matrix.

Otherwise returns only the loss.</li>

</ul>

</dd>

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

<li>loss (float) – Cost associated to the optimal transportation</li>

<li>log (dict) – If input log is True, a dictionary containing the Optimal transportation

matrix for the given parameters</li>

</ul>

</dd>

</dl>

Examples

Simple example with obvious solution. The function emd2_1d accepts lists and

performs automatic conversion to numpy arrays

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

>>> a=[.5, .5]

>>> b=[.5, .5]

>>> x_a = [2., 0.]

>>> x_b = [0., 3.]

>>> ot.emd2_1d(x_a, x_b, a, b)

0.5

</pre></div>

</div>

References

<dd>Peyré, G., & Cuturi, M. (2017). “Computational Optimal

Transport”, 2018.

</dd>

</dl>

See also

<dt><a class="reference internal" href="gen_modules/ot.lp.html#ot.lp.emd2" title="ot.lp.emd2"><code class="xref py py-func docutils literal notranslate">ot.lp.emd2()</code></a></dt><dd>EMD for multidimensional distributions

</dd>

<dt><a class="reference internal" href="gen_modules/ot.lp.html#ot.lp.emd_1d" title="ot.lp.emd_1d"><code class="xref py py-func docutils literal notranslate">ot.lp.emd_1d()</code></a></dt><dd>EMD for 1d distributions (returns the transportation matrix instead of the cost)

</dd>

</dl>

</div>

</dd></dl>

<code class="sig-prename descclassname">ot.</code><code class="sig-name descname">emd_1d</code>(x_a, x_b, a=None, b=None, metric='sqeuclidean', p=1.0, dense=True, log=False)<a class="reference internal" href="_modules/ot/lp.html#emd_1d">[source]</a><a class="headerlink" href="#ot.emd_1d" title="Permalink to this definition">¶</a></dt>

<dd>Solves the Earth Movers distance problem between 1d measures and returns

the OT matrix

\[ \begin{align}\begin{aligned}\gamma = arg\min_\gamma \sum_i \sum_j \gamma_{ij} d(x_a[i], x_b[j])\\s.t. \gamma 1 = a,

\gamma^T 1= b,

\gamma\geq 0\end{aligned}\end{align} \]</div>

where :

<li>d is the metric</li>

<li>x_a and x_b are the samples</li>

<li>a and b are the sample weights</li>

</ul>

When ‘minkowski’ is used as a metric, \(d(x, y) = |x - y|^p\).

Uses the algorithm detailed in <a href="#id43">[1]_</a>

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

<li>a ((ns,) ndarray, float64, optional) – Source histogram (default is uniform weight)</li>

<li>b ((nt,) ndarray, float64, optional) – Target histogram (default is uniform weight)</li>

Due to implementation details, this function runs faster when

<cite>‘sqeuclidean’</cite>, <cite>‘cityblock’</cite>, or <cite>‘euclidean’</cite> metrics are used.</li>

Otherwise returns a sparse representation using scipy’s <cite>coo_matrix</cite>

format. Due to implementation details, this function runs faster when

<cite>‘sqeuclidean’</cite>, <cite>‘minkowski’</cite>, <cite>‘cityblock’</cite>, or <cite>‘euclidean’</cite> metrics

are used.</li>

<li>log (boolean, optional (default=False)) – If True, returns a dictionary containing the cost.

Otherwise returns only the optimal transportation matrix.</li>

</ul>

</dd>

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

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

<li>log (dict) – If input log is True, a dictionary containing the cost</li>

</ul>

</dd>

</dl>

Examples

Simple example with obvious solution. The function emd_1d accepts lists and

performs automatic conversion to numpy arrays

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

>>> a=[.5, .5]

>>> b=[.5, .5]

>>> ot.emd_1d(x_a, x_b, a, b)

array([[0. , 0.5],

[0.5, 0. ]])

array([[0. , 0.5],

[0.5, 0. ]])

</pre></div>

</div>

References

<dd>Peyré, G., & Cuturi, M. (2017). “Computational Optimal

Transport”, 2018.

</dd>

</dl>

See also

<dt><a class="reference internal" href="gen_modules/ot.lp.html#ot.lp.emd" title="ot.lp.emd"><code class="xref py py-func docutils literal notranslate">ot.lp.emd()</code></a></dt><dd>EMD for multidimensional distributions

</dd>

<dt><a class="reference internal" href="gen_modules/ot.lp.html#ot.lp.emd2_1d" title="ot.lp.emd2_1d"><code class="xref py py-func docutils literal notranslate">ot.lp.emd2_1d()</code></a></dt><dd>EMD for 1d distributions (returns cost instead of the transportation matrix)

</dd>

</dl>

</div>

</dd></dl>

<code class="sig-prename descclassname">ot.</code><code class="sig-name descname">emd_1d</code>(x_a, x_b, a=None, b=None, metric='sqeuclidean', p=1.0, dense=True, log=False)<a class="reference internal" href="_modules/ot/lp.html#emd_1d">[source]</a><a class="headerlink" href="#id0" title="Permalink to this definition">¶</a></dt>

<dd>Solves the Earth Movers distance problem between 1d measures and returns

the OT matrix

\[ \begin{align}\begin{aligned}\gamma = arg\min_\gamma \sum_i \sum_j \gamma_{ij} d(x_a[i], x_b[j])\\s.t. \gamma 1 = a,

\gamma^T 1= b,

\gamma\geq 0\end{aligned}\end{align} \]</div>

where :

<li>d is the metric</li>

<li>x_a and x_b are the samples</li>

<li>a and b are the sample weights</li>

</ul>

When ‘minkowski’ is used as a metric, \(d(x, y) = |x - y|^p\).

Uses the algorithm detailed in <a href="#id44">[1]_</a>

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

<li>a ((ns,) ndarray, float64, optional) – Source histogram (default is uniform weight)</li>

<li>b ((nt,) ndarray, float64, optional) – Target histogram (default is uniform weight)</li>

Due to implementation details, this function runs faster when

<cite>‘sqeuclidean’</cite>, <cite>‘cityblock’</cite>, or <cite>‘euclidean’</cite> metrics are used.</li>

Otherwise returns a sparse representation using scipy’s <cite>coo_matrix</cite>

format. Due to implementation details, this function runs faster when

<cite>‘sqeuclidean’</cite>, <cite>‘minkowski’</cite>, <cite>‘cityblock’</cite>, or <cite>‘euclidean’</cite> metrics

are used.</li>

<li>log (boolean, optional (default=False)) – If True, returns a dictionary containing the cost.

Otherwise returns only the optimal transportation matrix.</li>

</ul>

</dd>

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

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

<li>log (dict) – If input log is True, a dictionary containing the cost</li>

</ul>

</dd>

</dl>

Examples

Simple example with obvious solution. The function emd_1d accepts lists and

performs automatic conversion to numpy arrays

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

>>> a=[.5, .5]

>>> b=[.5, .5]

array([[0. , 0.5],

[0.5, 0. ]])

array([[0. , 0.5],

[0.5, 0. ]])

</pre></div>

</div>

References

<dd>Peyré, G., & Cuturi, M. (2017). “Computational Optimal

Transport”, 2018.

</dd>

</dl>

See also

</dd>

</dl>

</div>

</dd></dl>

<code class="sig-prename descclassname">ot.</code><code class="sig-name descname">sinkhorn</code>(a, b, M, reg, method='sinkhorn', numItermax=1000, stopThr=1e-09, verbose=False, log=False, **kwargs)<a class="reference internal" href="_modules/ot/bregman.html#sinkhorn">[source]</a><a class="headerlink" href="#ot.sinkhorn" title="Permalink to this definition">¶</a></dt>

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

The function solves the following optimization problem:

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

where :

<li>M is the (dim_a, dim_b) 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>a and b are source and target weights (histograms, both sum to 1)</li>

</ul>

The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in <a href="#id45">[2]_</a>

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

<li>a (ndarray, shape (dim_a,)) – samples weights in the source domain</li>

<li>b (ndarray, shape (dim_b,) or ndarray, shape (dim_b, n_hists)) – samples in the target domain, compute sinkhorn with multiple targets

and fixed M if b is a matrix (return OT loss + dual variables in log)</li>

<li>M (ndarray, shape (dim_a, dim_b)) – loss matrix</li>

<li>method (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.8)">str</a>) – method used for the solver either ‘sinkhorn’, ‘greenkhorn’, ‘sinkhorn_stabilized’ or

‘sinkhorn_epsilon_scaling’, see those function for specific parameters</li>

</ul>

</dd>

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

<li>gamma (ndarray, shape (dim_a, dim_b)) – 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 ot

>>> a=[.5, .5]

>>> b=[.5, .5]

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

>>> ot.sinkhorn(a, b, M, 1)

array([[0.36552929, 0.13447071],

[0.13447071, 0.36552929]])

</pre></div>

</div>

References

<li>Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport, Advances in Neural Information Processing Systems (NIPS) 26, 2013</li>

</ol>

</dd>

<dd>Schmitzer, B. (2016). Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems. arXiv preprint arXiv:1610.06519.

</dd>

<dd>Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.

</dd>

</dl>

See also

</dd>

<dt><a class="reference internal" href="gen_modules/ot.bregman.html#ot.bregman.sinkhorn_knopp" title="ot.bregman.sinkhorn_knopp"><code class="xref py py-func docutils literal notranslate">ot.bregman.sinkhorn_knopp()</code></a></dt><dd>Classic Sinkhorn [2]

</dd>

<dt><a class="reference internal" href="gen_modules/ot.bregman.html#ot.bregman.sinkhorn_stabilized" title="ot.bregman.sinkhorn_stabilized"><code class="xref py py-func docutils literal notranslate">ot.bregman.sinkhorn_stabilized()</code></a></dt><dd>Stabilized sinkhorn [9][10]

</dd>

<dt><a class="reference internal" href="gen_modules/ot.bregman.html#ot.bregman.sinkhorn_epsilon_scaling" title="ot.bregman.sinkhorn_epsilon_scaling"><code class="xref py py-func docutils literal notranslate">ot.bregman.sinkhorn_epsilon_scaling()</code></a></dt><dd>Sinkhorn with epslilon scaling [9][10]

</dd>

</dl>

</div>

</dd></dl>

<code class="sig-prename descclassname">ot.</code><code class="sig-name descname">sinkhorn2</code>(a, b, M, reg, method='sinkhorn', numItermax=1000, stopThr=1e-09, verbose=False, log=False, **kwargs)<a class="reference internal" href="_modules/ot/bregman.html#sinkhorn2">[source]</a><a class="headerlink" href="#ot.sinkhorn2" title="Permalink to this definition">¶</a></dt>

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

The function solves the following optimization problem:

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

where :

<li>M is the (dim_a, dim_b) metric cost matrix</li>

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

</ul>

The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in <a href="#id46">[2]_</a>

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

<li>a (ndarray, shape (dim_a,)) – samples weights in the source domain</li>

and fixed M if b is a matrix (return OT loss + dual variables in log)</li>

<li>M (ndarray, shape (dim_a, dim_b)) – loss matrix</li>

<li>method (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.8)">str</a>) – method used for the solver either ‘sinkhorn’, ‘sinkhorn_stabilized’ or

‘sinkhorn_epsilon_scaling’, see those function for specific parameters</li>

</ul>

</dd>

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

<li>W ((n_hists) ndarray 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

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

>>> a=[.5, .5]

>>> b=[.5, .5]

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

>>> ot.sinkhorn2(a, b, M, 1)

array([0.26894142])

</pre></div>

</div>

References

<li>Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport, Advances in Neural Information Processing Systems (NIPS) 26, 2013</li>

</ol>

</dd>

<dd>Schmitzer, B. (2016). Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems. arXiv preprint arXiv:1610.06519.

</dd>

<dd>Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.

[21] Altschuler J., Weed J., Rigollet P. : Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration, Advances in Neural Information Processing Systems (NIPS) 31, 2017

</dd>

</dl>

See also

</dd>

<dt><a class="reference internal" href="gen_modules/ot.bregman.html#ot.bregman.greenkhorn" title="ot.bregman.greenkhorn"><code class="xref py py-func docutils literal notranslate">ot.bregman.greenkhorn()</code></a></dt><dd>Greenkhorn [21]

</dd>

</dl>

</div>

</dd></dl>

<code class="sig-prename descclassname">ot.</code><code class="sig-name descname">sinkhorn_lpl1_mm</code>(a, labels_a, b, M, reg, eta=0.1, numItermax=10, numInnerItermax=200, stopInnerThr=1e-09, verbose=False, log=False)<a class="reference internal" href="_modules/ot/da.html#sinkhorn_lpl1_mm">[source]</a><a class="headerlink" href="#ot.sinkhorn_lpl1_mm" title="Permalink to this definition">¶</a></dt>

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

group lasso regularization

The function solves the following optimization problem:

\[ \begin{align}\begin{aligned}\gamma = arg\min_\gamma <\gamma,M>_F + reg\cdot\Omega_e(\gamma)

+ \eta \Omega_g(\gamma)\\s.t. \gamma 1 = a\\ \gamma^T 1= b\\ \gamma\geq 0\end{aligned}\end{align} \]</div>

where :

<li>M is the (ns,nt) metric cost matrix</li>

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

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

<li>\(\Omega_g\) is the group lasso regularization term

\(\Omega_g(\gamma)=\sum_{i,c} \|\gamma_{i,\mathcal{I}_c}\|^{1/2}_1\)

where \(\mathcal{I}_c\) are the index of samples from class c

in the source domain.</li>

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

</ul>

The algorithm used for solving the problem is the generalized conditional

gradient as proposed in <a class="footnote-reference brackets" href="#id26" id="id24">5</a> <a class="footnote-reference brackets" href="#id27" id="id25">7</a>

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

<li>a (np.ndarray (ns,)) – samples weights in the source domain</li>

<li>labels_a (np.ndarray (ns,)) – labels of samples in the source domain</li>

<li>b (np.ndarray (nt,)) – samples weights in the target domain</li>

<li>M (np.ndarray (ns,nt)) – loss matrix</li>

<li>eta (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.8)">float</a>, optional) – Regularization term for group lasso regularization >0</li>

<li>numInnerItermax (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.8)">int</a>, optional) – Max number of iterations (inner sinkhorn solver)</li>

<li>stopInnerThr (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.8)">float</a>, optional) – Stop threshold on error (inner sinkhorn solver) (>0)</li>

</ul>

</dd>

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

<li>gamma ((ns x nt) ndarray) – Optimal transportation matrix for the given parameters</li>

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

</ul>

</dd>

</dl>

References

<dd>N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy,

“Optimal Transport for Domain Adaptation,” in IEEE

Transactions on Pattern Analysis and Machine Intelligence ,

vol.PP, no.99, pp.1-1

</dd>

<dd>Rakotomamonjy, A., Flamary, R., & Courty, N. (2015).

Generalized conditional gradient: analysis of convergence

and applications. arXiv preprint arXiv:1510.06567.

</dd>

</dl>

See also

</dd>

</dl>

</div>

</dd></dl>

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