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<!DOCTYPE html>

<head>

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

</head>

POT Python Optimal Transport

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<li class="toctree-l3"><a class="reference internal" href="#functions">Functions</a><ul>

<li class="toctree-l4"><a class="reference internal" href="#ot.mapping.joint_OT_mapping_kernel"><code class="docutils literal notranslate">joint_OT_mapping_kernel()</code></a></li>

<li class="toctree-l4"><a class="reference internal" href="#ot.mapping.joint_OT_mapping_linear"><code class="docutils literal notranslate">joint_OT_mapping_linear()</code></a></li>

<li class="toctree-l4"><a class="reference internal" href="#ot.mapping.nearest_brenier_potential_fit"><code class="docutils literal notranslate">nearest_brenier_potential_fit()</code></a></li>

<li class="toctree-l4"><a class="reference internal" href="#examples-using-ot-mapping-nearest-brenier-potential-fit">Examples using <code class="docutils literal notranslate">ot.mapping.nearest_brenier_potential_fit</code></a></li>

<li class="toctree-l4"><a class="reference internal" href="#examples-using-ot-mapping-nearest-brenier-potential-predict-bounds">Examples using <code class="docutils literal notranslate">ot.mapping.nearest_brenier_potential_predict_bounds</code></a></li>

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<li class="toctree-l3"><a class="reference internal" href="#id0"><code class="docutils literal notranslate">joint_OT_mapping_kernel()</code></a></li>

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<li class="toctree-l3"><a class="reference internal" href="#id11"><code class="docutils literal notranslate">nearest_brenier_potential_fit()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#id13"><code class="docutils literal notranslate">nearest_brenier_potential_predict_bounds()</code></a></li>

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

Optimal Transport maps and variants

Warning

Note that by default the module is not imported in <a class="reference internal" href="../all.html#module-ot" title="ot"><code class="xref py py-mod docutils literal notranslate">ot</code></a>. In order to

use it you need to explicitly import <a class="reference internal" href="#module-ot.mapping" title="ot.mapping"><code class="xref py py-mod docutils literal notranslate">ot.mapping</code></a>

</div>

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

ot.mapping.joint_OT_mapping_kernel(xs, xt, mu=1, eta=0.001, kerneltype='gaussian', sigma=1, bias=False, verbose=False, verbose2=False, numItermax=100, numInnerItermax=10, stopInnerThr=1e-06, stopThr=1e-05, log=False, **kwargs)<a class="reference internal" href="../_modules/ot/mapping.html#joint_OT_mapping_kernel">[source]</a><a class="headerlink" href="#ot.mapping.joint_OT_mapping_kernel" title="Link to this definition"></a></dt>

<dd>Joint OT and nonlinear mapping estimation with kernels as proposed in

[8].

The function solves the following optimization problem:

\[ \begin{align}\begin{aligned}\min_{\gamma, L\in\mathcal{H}}\quad \|L(\mathbf{X_s}) -

n_s\gamma \mathbf{X_t}\|^2_F + \mu \langle \gamma, \mathbf{M} \rangle_F +

\eta \|L\|^2_\mathcal{H}\\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>ns</cite>, <cite>nt</cite>) squared euclidean cost matrix between samples in

\(\mathbf{X_s}\) and \(\mathbf{X_t}\) (scaled by \(n_s\))</li>

<li>\(L\) is a \(n_s \times d\) linear operator on a kernel matrix that

approximates the barycentric mapping</li>

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

</ul>

The problem consist in solving jointly an optimal transport matrix

\(\gamma\) and the nonlinear mapping that fits the barycentric mapping

\(n_s\gamma \mathbf{X_t}\).

One can also estimate a mapping with constant bias (see supplementary

material of [8]) using the bias optional argument.

The algorithm used for solving the problem is the block coordinate

descent that alternates between updates of \(\mathbf{G}\) (using conditional gradient)

and the update of \(\mathbf{L}\) using a classical kernel least square solver.

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

<li>xs (array-like (ns,d)) – samples in the source domain</li>

<li>xt (array-like (nt,d)) – samples in the target domain</li>

<li>mu (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>,optional) – Weight for the linear OT loss (>0)</li>

<li>eta (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>, optional) – Regularization term for the linear mapping L (>0)</li>

<li>kerneltype (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)">str</a>,optional) – kernel used by calling function <a class="reference internal" href="ot.utils.html#id104" title="ot.utils.kernel"><code class="xref py py-func docutils literal notranslate">ot.utils.kernel()</code></a> (gaussian by default)</li>

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

<li>bias (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#bool" title="(in Python v3.15)">bool</a>,optional) – Estimate linear mapping with constant bias</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>

<li>verbose2 (<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>

<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 BCD iterations</li>

<li>numInnerItermax (<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 (inner CG solver)</li>

<li>stopInnerThr (<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 (inner CG solver) (>0)</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 relative loss decrease (>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) – record log if True</li>

</ul>

</dd>

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

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

<li>L ((ns, d) array-like) – Nonlinear mapping matrix ((\(n_s+1\), <cite>d</cite>) if bias)</li>

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

</ul>

</dd>

</dl>

References

[8]

M. Perrot, N. Courty, R. Flamary, A. Habrard,

“Mapping estimation for discrete optimal transport”,

Neural Information Processing Systems (NIPS), 2016.

</aside>

See also

<dt><a class="reference internal" href="ot.lp.html#ot.lp.emd" title="ot.lp.emd"><code class="xref py py-obj docutils literal notranslate">ot.lp.emd</code></a></dt><dd>Unregularized OT

</dd>

<dt><a class="reference internal" href="ot.optim.html#id0" title="ot.optim.cg"><code class="xref py py-obj docutils literal notranslate">ot.optim.cg</code></a></dt><dd>General regularized OT

</dd>

</dl>

</div>

</dd></dl>

ot.mapping.joint_OT_mapping_linear(xs, xt, mu=1, eta=0.001, bias=False, verbose=False, verbose2=False, numItermax=100, numInnerItermax=10, stopInnerThr=1e-06, stopThr=1e-05, log=False, **kwargs)<a class="reference internal" href="../_modules/ot/mapping.html#joint_OT_mapping_linear">[source]</a><a class="headerlink" href="#ot.mapping.joint_OT_mapping_linear" title="Link to this definition"></a></dt>

<dd>Joint OT and linear mapping estimation as proposed in

[8].

The function solves the following optimization problem:

\[ \begin{align}\begin{aligned}\min_{\gamma,L}\quad \|L(\mathbf{X_s}) - n_s\gamma \mathbf{X_t} \|^2_F +

\mu \langle \gamma, \mathbf{M} \rangle_F + \eta \|L - \mathbf{I}\|^2_F\\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>ns</cite>, <cite>nt</cite>) squared euclidean cost matrix between samples in

<li>\(L\) is a \(d\times d\) linear operator that approximates the barycentric

mapping</li>

<li>\(\mathbf{I}\) is the identity matrix (neutral linear mapping)</li>

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

</ul>

The problem consist in solving jointly an optimal transport matrix

\(\gamma\) and a linear mapping that fits the barycentric mapping

\(n_s\gamma \mathbf{X_t}\).

One can also estimate a mapping with constant bias (see supplementary

material of [8]) using the bias optional argument.

The algorithm used for solving the problem is the block coordinate

descent that alternates between updates of \(\mathbf{G}\) (using conditional gradient)

and the update of \(\mathbf{L}\) using a classical least square solver.

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

<li>xs (array-like (ns,d)) – samples in the source domain</li>

<li>xt (array-like (nt,d)) – samples in the target domain</li>

</ul>

</dd>

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

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

<li>L ((d, d) array-like) – Linear mapping matrix ((\(d+1\), <cite>d</cite>) if bias)</li>

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

</ul>

</dd>

</dl>

References

[8]

M. Perrot, N. Courty, R. Flamary, A. Habrard,

“Mapping estimation for discrete optimal transport”,

Neural Information Processing Systems (NIPS), 2016.

</aside>

See also

</dd>

</dl>

</div>

</dd></dl>

ot.mapping.nearest_brenier_potential_fit(X, V, X_classes=None, a=None, b=None, strongly_convex_constant=0.6, gradient_lipschitz_constant=1.4, its=100, log=False, init_method='barycentric', solver=None)<a class="reference internal" href="../_modules/ot/mapping.html#nearest_brenier_potential_fit">[source]</a><a class="headerlink" href="#ot.mapping.nearest_brenier_potential_fit" title="Link to this definition"></a></dt>

<dd>Computes optimal values and gradients at X for a strongly convex potential \(\varphi\) with Lipschitz gradients

on the partitions defined by <cite>X_classes</cite>, where \(\varphi\) is optimal such that

\(\nabla \varphi \#\mu \approx \nu\), given samples \(X = x_1, \cdots, x_n \sim \mu\) and

\(V = v_1, \cdots, v_n \sim \nu\). Finding such a potential that has the desired regularity on the

partition \((E_k)_{k \in [K]}\) (given by the classes <cite>X_classes</cite>) is equivalent to finding optimal values

<cite>phi</cite> for the \(\varphi(x_i)\) and its gradients \(\nabla \varphi(x_i)\) (variable`G`).

In practice, these optimal values are found by solving the following problem

\begin{gather*}

\text{min} \sum_{i,j}\pi_{i,j}\|g_i - v_j\|_2^2 \\

g_1,\cdots, g_n \in \mathbb{R}^d,\; \varphi_1, \cdots, \varphi_n \in \mathbb{R},\; \pi \in \Pi(a, b) \\

\text{s.t.}\ \forall k \in [K],\; \forall i,j \in I_k: \\

\varphi_i-\varphi_j-\langle g_j, x_i-x_j\rangle \geq c_1\|g_i - g_j\|_2^2

+ c_2\|x_i-x_j\|_2^2 - c_3\langle g_j-g_i, x_j -x_i \rangle.

\end{gather*}</div>The constants \(c_1, c_2, c_3\) only depend on <cite>strongly_convex_constant</cite> and <cite>gradient_lipschitz_constant</cite>.

The constraint \(\pi \in \Pi(a, b)\) denotes the fact that the matrix \(\pi\) belong to the OT polytope

of marginals a and b. \(I_k\) is the subset of \([n]\) of the i such that \(x_i\) is in the

partition (or class) \(E_k\), i.e. <cite>X_classes[i] == k</cite>.

This problem is solved by alternating over the variable \(\pi\) and the variables \(\varphi_i, g_i\).

For \(\pi\), the problem is the standard discrete OT problem, and for \(\varphi_i, g_i\), the

problem is a convex QCQP solved using <code class="code docutils literal notranslate">cvxpy</code> (ECOS solver).

Accepts any compatible backend, but will perform the QCQP optimisation on Numpy arrays, and convert back at the end.

Warning

This function requires the CVXPY library

</div>

Warning

Accepts any backend but will convert to Numpy then back to the backend.

</div>

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

<li>X (array-like (n, d)) – reference points used to compute the optimal values phi and G</li>

<li>V (array-like (n, d)) – values of the gradients at the reference points X</li>

<li>X_classes (array-like (n,), optional) – classes of the reference points, defaults to a single class</li>

<li>a (array-like (n,), optional) – weights for the reference points X, defaults to uniform</li>

<li>b (array-like (n,), optional) – weights for the target points V, defaults to uniform</li>

<li>strongly_convex_constant (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>, optional) – constant for the strong convexity of the input potential phi, defaults to 0.6</li>

<li>gradient_lipschitz_constant (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>, optional) – constant for the Lipschitz property of the input gradient G, defaults to 1.4</li>

<li>its (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional) – number of iterations, defaults to 100</li>

<li>init_method (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)">str</a>, optional) – ‘target’ initialises G=V, ‘barycentric’ initialises at the image of X by

the barycentric projection</li>

<li>solver (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)">str</a>, optional) – The CVXPY solver to use</li>

</ul>

</dd>

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

<li>phi (array-like (n,)) – optimal values of the potential at the points X</li>

<li>G (array-like (n, d)) – optimal values of the gradients at the points X</li>

<li>log (dict, optional) – If input log is true, a dictionary containing the values of the variables at each iteration, as well

as solver information</li>

</ul>

</dd>

</dl>

References

[58]

François-Pierre Paty, Alexandre d’Aspremont, and Marco Cuturi. Regularity as regularization:

Smooth and strongly convex brenier potentials in optimal transport. In International Conference

on Artificial Intelligence and Statistics, pages 1222–1232. PMLR, 2020.

</aside>

See also

<dt><a class="reference internal" href="#id13" title="ot.mapping.nearest_brenier_potential_predict_bounds"><code class="xref py py-obj docutils literal notranslate">ot.mapping.nearest_brenier_potential_predict_bounds</code></a></dt><dd>Predicting SSNB images on new source data

</dd>

<dt><a class="reference internal" href="ot.da.html#id90" title="ot.da.NearestBrenierPotential"><code class="xref py py-obj docutils literal notranslate">ot.da.NearestBrenierPotential</code></a></dt><dd>BaseTransport wrapper for SSNB

</dd>

</dl>

</div>

</dd></dl>

<h3>Examples using <code class="docutils literal notranslate">ot.mapping.nearest_brenier_potential_fit</code><a class="headerlink" href="#examples-using-ot-mapping-nearest-brenier-potential-fit" title="Link to this heading"></a></h3>

<a class="reference internal" href="../auto_examples/others/plot_SSNB.html">Smooth and Strongly Convex Nearest Brenier Potentials</a>

<div class="sphx-glr-thumbnail-title">Smooth and Strongly Convex Nearest Brenier Potentials</div>

</div></div><div class="sphx-glr-clear"></div><dl class="py function">

ot.mapping.nearest_brenier_potential_predict_bounds(X, phi, G, Y, X_classes=None, Y_classes=None, strongly_convex_constant=0.6, gradient_lipschitz_constant=1.4, log=False, solver=None)<a class="reference internal" href="../_modules/ot/mapping.html#nearest_brenier_potential_predict_bounds">[source]</a><a class="headerlink" href="#ot.mapping.nearest_brenier_potential_predict_bounds" title="Link to this definition"></a></dt>

<dd>Compute the values of the lower and upper bounding potentials at the input points Y, using the potential optimal

values phi at X and their gradients G at X. The ‘lower’ potential corresponds to the method from [58],

Equation 2, while the bounding property and ‘upper’ potential come from [59], Theorem 3.14 (taking into

account the fact that this theorem’s statement has a min instead of a max, which is a typo). Both potentials are

optimal for the SSNB problem.

If \(I_k\) is the subset of \([n]\) of the i such that \(x_i\) is in the partition (or class)

\(E_k\), for each \(y \in E_k\), this function solves the convex QCQP problems,

respectively for l: ‘lower’ and u: ‘upper’:

\begin{gather*}

(\varphi_{l}(x), \nabla \varphi_l(x)) = \text{argmin}\ t, \\

t\in \mathbb{R},\; g\in \mathbb{R}^d, \\

\text{s.t.} \forall j \in I_k,\; t-\varphi_j - \langle g_j, y-x_j \rangle \geq c_1\|g - g_j\|_2^2

+ c_2\|y-x_j\|_2^2 - c_3\langle g_j-g, x_j -y \rangle.

\end{gather*}</div><div class="math notranslate nohighlight">

\begin{gather*}

(\varphi_{u}(x), \nabla \varphi_u(x)) = \text{argmax}\ t, \\

t\in \mathbb{R},\; g\in \mathbb{R}^d, \\

\text{s.t.} \forall i \in I_k,\; \varphi_i^* -t - \langle g, x_i-y \rangle \geq c_1\|g_i - g\|_2^2

+ c_2\|x_i-y\|_2^2 - c_3\langle g-g_i, y -x_i \rangle.

Warning

This function requires the CVXPY library

</div>

Warning

Accepts any backend but will convert to Numpy then back to the backend.

</div>

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

<li>X (array-like (n, d)) – reference points used to compute the optimal values phi and G</li>

<li>phi (array-like (n,)) – optimal values of the potential at the points X</li>

<li>G (array-like (n, d)) – optimal values of the gradients at the points X</li>

<li>Y (array-like (m, d)) – input points</li>

<li>X_classes (array-like (n,), optional) – classes of the reference points, defaults to a single class</li>

<li>Y_classes (array_like (m,), optional) – classes of the input points, defaults to a single class</li>

</ul>

</dd>

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

<li>phi_lu (array-like (2, m)) – values of the lower and upper bounding potentials at Y</li>

<li>G_lu (array-like (2, m, d)) – gradients of the lower and upper bounding potentials at Y</li>

<li>log (dict, optional) – If input log is true, a dictionary containing solver information</li>

</ul>

</dd>

</dl>

References

[58]

François-Pierre Paty, Alexandre d’Aspremont, and Marco Cuturi. Regularity as regularization:

Smooth and strongly convex brenier potentials in optimal transport. In International Conference

on Artificial Intelligence and Statistics, pages 1222–1232. PMLR, 2020.

</aside>

[59]

Adrien B Taylor. Convex interpolation and performance estimation of first-order methods for

convex optimization. PhD thesis, Catholic University of Louvain, Louvain-la-Neuve, Belgium,

2017.

</aside>

See also

<dt><a class="reference internal" href="#id11" title="ot.mapping.nearest_brenier_potential_fit"><code class="xref py py-obj docutils literal notranslate">ot.mapping.nearest_brenier_potential_fit</code></a></dt><dd>Fitting the SSNB on source and target data

</dd>

</dl>

</div>

</dd></dl>

</section>

<h3>Examples using <code class="docutils literal notranslate">ot.mapping.nearest_brenier_potential_predict_bounds</code><a class="headerlink" href="#examples-using-ot-mapping-nearest-brenier-potential-predict-bounds" title="Link to this heading"></a></h3>

<a class="reference internal" href="../auto_examples/others/plot_SSNB.html">Smooth and Strongly Convex Nearest Brenier Potentials</a>

<div class="sphx-glr-thumbnail-title">Smooth and Strongly Convex Nearest Brenier Potentials</div>

</div></div><div class="sphx-glr-clear"></div></section>

</section>

<dd>Joint OT and nonlinear mapping estimation with kernels as proposed in

[8].

The function solves the following optimization problem:

\[ \begin{align}\begin{aligned}\min_{\gamma, L\in\mathcal{H}}\quad \|L(\mathbf{X_s}) -

n_s\gamma \mathbf{X_t}\|^2_F + \mu \langle \gamma, \mathbf{M} \rangle_F +

\eta \|L\|^2_\mathcal{H}\\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>ns</cite>, <cite>nt</cite>) squared euclidean cost matrix between samples in

<li>\(L\) is a \(n_s \times d\) linear operator on a kernel matrix that

approximates the barycentric mapping</li>

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

</ul>

The problem consist in solving jointly an optimal transport matrix

\(\gamma\) and the nonlinear mapping that fits the barycentric mapping

\(n_s\gamma \mathbf{X_t}\).

One can also estimate a mapping with constant bias (see supplementary

material of [8]) using the bias optional argument.

The algorithm used for solving the problem is the block coordinate

descent that alternates between updates of \(\mathbf{G}\) (using conditional gradient)

and the update of \(\mathbf{L}\) using a classical kernel least square solver.

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

<li>xs (array-like (ns,d)) – samples in the source domain</li>

<li>xt (array-like (nt,d)) – samples in the target domain</li>

</ul>

</dd>

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

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

<li>L ((ns, d) array-like) – Nonlinear mapping matrix ((\(n_s+1\), <cite>d</cite>) if bias)</li>

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

</ul>

</dd>

</dl>

References

[8]

M. Perrot, N. Courty, R. Flamary, A. Habrard,

“Mapping estimation for discrete optimal transport”,

Neural Information Processing Systems (NIPS), 2016.

</aside>

See also

</dd>

</dl>

</div>

</dd></dl>

<dd>Joint OT and linear mapping estimation as proposed in

[8].

The function solves the following optimization problem:

\[ \begin{align}\begin{aligned}\min_{\gamma,L}\quad \|L(\mathbf{X_s}) - n_s\gamma \mathbf{X_t} \|^2_F +

\mu \langle \gamma, \mathbf{M} \rangle_F + \eta \|L - \mathbf{I}\|^2_F\\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>ns</cite>, <cite>nt</cite>) squared euclidean cost matrix between samples in

<li>\(L\) is a \(d\times d\) linear operator that approximates the barycentric

mapping</li>

<li>\(\mathbf{I}\) is the identity matrix (neutral linear mapping)</li>

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

</ul>

The problem consist in solving jointly an optimal transport matrix

\(\gamma\) and a linear mapping that fits the barycentric mapping

\(n_s\gamma \mathbf{X_t}\).

One can also estimate a mapping with constant bias (see supplementary

material of [8]) using the bias optional argument.

The algorithm used for solving the problem is the block coordinate

descent that alternates between updates of \(\mathbf{G}\) (using conditional gradient)

and the update of \(\mathbf{L}\) using a classical least square solver.

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

<li>xs (array-like (ns,d)) – samples in the source domain</li>

<li>xt (array-like (nt,d)) – samples in the target domain</li>

</ul>

</dd>

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

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

<li>L ((d, d) array-like) – Linear mapping matrix ((\(d+1\), <cite>d</cite>) if bias)</li>

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

</ul>

</dd>

</dl>

References

[8]

M. Perrot, N. Courty, R. Flamary, A. Habrard,

“Mapping estimation for discrete optimal transport”,

Neural Information Processing Systems (NIPS), 2016.

</aside>

See also

</dd>

</dl>

</div>

</dd></dl>

<dd>Computes optimal values and gradients at X for a strongly convex potential \(\varphi\) with Lipschitz gradients

on the partitions defined by <cite>X_classes</cite>, where \(\varphi\) is optimal such that

\(\nabla \varphi \#\mu \approx \nu\), given samples \(X = x_1, \cdots, x_n \sim \mu\) and

\(V = v_1, \cdots, v_n \sim \nu\). Finding such a potential that has the desired regularity on the

partition \((E_k)_{k \in [K]}\) (given by the classes <cite>X_classes</cite>) is equivalent to finding optimal values

<cite>phi</cite> for the \(\varphi(x_i)\) and its gradients \(\nabla \varphi(x_i)\) (variable`G`).

In practice, these optimal values are found by solving the following problem

\begin{gather*}

\text{min} \sum_{i,j}\pi_{i,j}\|g_i - v_j\|_2^2 \\

g_1,\cdots, g_n \in \mathbb{R}^d,\; \varphi_1, \cdots, \varphi_n \in \mathbb{R},\; \pi \in \Pi(a, b) \\

\text{s.t.}\ \forall k \in [K],\; \forall i,j \in I_k: \\

\varphi_i-\varphi_j-\langle g_j, x_i-x_j\rangle \geq c_1\|g_i - g_j\|_2^2

+ c_2\|x_i-x_j\|_2^2 - c_3\langle g_j-g_i, x_j -x_i \rangle.

partition (or class) \(E_k\), i.e. <cite>X_classes[i] == k</cite>.

For \(\pi\), the problem is the standard discrete OT problem, and for \(\varphi_i, g_i\), the

problem is a convex QCQP solved using <code class="code docutils literal notranslate">cvxpy</code> (ECOS solver).

Accepts any compatible backend, but will perform the QCQP optimisation on Numpy arrays, and convert back at the end.

Warning

This function requires the CVXPY library

</div>

Warning

Accepts any backend but will convert to Numpy then back to the backend.

</div>

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

<li>X (array-like (n, d)) – reference points used to compute the optimal values phi and G</li>

<li>V (array-like (n, d)) – values of the gradients at the reference points X</li>

<li>X_classes (array-like (n,), optional) – classes of the reference points, defaults to a single class</li>

<li>a (array-like (n,), optional) – weights for the reference points X, defaults to uniform</li>

<li>b (array-like (n,), optional) – weights for the target points V, defaults to uniform</li>

the barycentric projection</li>

</ul>

</dd>

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

<li>phi (array-like (n,)) – optimal values of the potential at the points X</li>

<li>G (array-like (n, d)) – optimal values of the gradients at the points X</li>

<li>log (dict, optional) – If input log is true, a dictionary containing the values of the variables at each iteration, as well

as solver information</li>

</ul>

</dd>

</dl>

References

[58]

François-Pierre Paty, Alexandre d’Aspremont, and Marco Cuturi. Regularity as regularization:

Smooth and strongly convex brenier potentials in optimal transport. In International Conference

on Artificial Intelligence and Statistics, pages 1222–1232. PMLR, 2020.

</aside>

See also

</dd>

</dl>

</div>

</dd></dl>

ot.mapping.nearest_brenier_potential_predict_bounds(X, phi, G, Y, X_classes=None, Y_classes=None, strongly_convex_constant=0.6, gradient_lipschitz_constant=1.4, log=False, solver=None)<a class="reference internal" href="../_modules/ot/mapping.html#nearest_brenier_potential_predict_bounds">[source]</a><a class="headerlink" href="#id13" title="Link to this definition"></a></dt>

<dd>Compute the values of the lower and upper bounding potentials at the input points Y, using the potential optimal

values phi at X and their gradients G at X. The ‘lower’ potential corresponds to the method from [58],

Equation 2, while the bounding property and ‘upper’ potential come from [59], Theorem 3.14 (taking into

account the fact that this theorem’s statement has a min instead of a max, which is a typo). Both potentials are

optimal for the SSNB problem.

\(E_k\), for each \(y \in E_k\), this function solves the convex QCQP problems,

respectively for l: ‘lower’ and u: ‘upper’:

\begin{gather*}

(\varphi_{l}(x), \nabla \varphi_l(x)) = \text{argmin}\ t, \\

t\in \mathbb{R},\; g\in \mathbb{R}^d, \\

\text{s.t.} \forall j \in I_k,\; t-\varphi_j - \langle g_j, y-x_j \rangle \geq c_1\|g - g_j\|_2^2

+ c_2\|y-x_j\|_2^2 - c_3\langle g_j-g, x_j -y \rangle.

\end{gather*}</div><div class="math notranslate nohighlight">

\begin{gather*}

(\varphi_{u}(x), \nabla \varphi_u(x)) = \text{argmax}\ t, \\

t\in \mathbb{R},\; g\in \mathbb{R}^d, \\

\text{s.t.} \forall i \in I_k,\; \varphi_i^* -t - \langle g, x_i-y \rangle \geq c_1\|g_i - g\|_2^2

+ c_2\|x_i-y\|_2^2 - c_3\langle g-g_i, y -x_i \rangle.

Warning

This function requires the CVXPY library

</div>

Warning

Accepts any backend but will convert to Numpy then back to the backend.

</div>

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

<li>X (array-like (n, d)) – reference points used to compute the optimal values phi and G</li>

<li>phi (array-like (n,)) – optimal values of the potential at the points X</li>

<li>G (array-like (n, d)) – optimal values of the gradients at the points X</li>

<li>Y (array-like (m, d)) – input points</li>

<li>X_classes (array-like (n,), optional) – classes of the reference points, defaults to a single class</li>

<li>Y_classes (array_like (m,), optional) – classes of the input points, defaults to a single class</li>

</ul>

</dd>

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

<li>phi_lu (array-like (2, m)) – values of the lower and upper bounding potentials at Y</li>

<li>G_lu (array-like (2, m, d)) – gradients of the lower and upper bounding potentials at Y</li>

<li>log (dict, optional) – If input log is true, a dictionary containing solver information</li>

</ul>

</dd>

</dl>

References

[58]

François-Pierre Paty, Alexandre d’Aspremont, and Marco Cuturi. Regularity as regularization:

Smooth and strongly convex brenier potentials in optimal transport. In International Conference

on Artificial Intelligence and Statistics, pages 1222–1232. PMLR, 2020.

</aside>

[59]

Adrien B Taylor. Convex interpolation and performance estimation of first-order methods for

convex optimization. PhD thesis, Catholic University of Louvain, Louvain-la-Neuve, Belgium,

2017.

</aside>

See also

</dd>

</dl>

</div>

</dd></dl>

</section>

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