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<!DOCTYPE html>

<head>

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

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POT Python Optimal Transport

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

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

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

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

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

<li class="toctree-l4"><a class="reference internal" href="#examples-using-ot-regpath-compute-transport-plan">Examples using <code class="docutils literal notranslate">ot.regpath.compute_transport_plan</code></a></li>

<li class="toctree-l4"><a class="reference internal" href="#examples-using-ot-regpath-regularization-path">Examples using <code class="docutils literal notranslate">ot.regpath.regularization_path</code></a></li>

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

<li class="toctree-l3"><a class="reference internal" href="#id12"><code class="docutils literal notranslate">compute_next_removal()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#id15"><code class="docutils literal notranslate">compute_transport_plan()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#id18"><code class="docutils literal notranslate">construct_augmented_H()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#id19"><code class="docutils literal notranslate">fully_relaxed_path()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#id21"><code class="docutils literal notranslate">ot_next_gamma()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#id23"><code class="docutils literal notranslate">recast_ot_as_lasso()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#id25"><code class="docutils literal notranslate">recast_semi_relaxed_as_lasso()</code></a></li>

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<li class="toctree-l3"><a class="reference internal" href="#id30"><code class="docutils literal notranslate">semi_relaxed_path()</code></a></li>

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

Regularization path OT solvers

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

ot.regpath.complement_schur(M_current, b, d, id_pop)<a class="reference internal" href="../_modules/ot/regpath.html#complement_schur">[source]</a><a class="headerlink" href="#ot.regpath.complement_schur" title="Link to this definition"></a></dt>

<dd>This function computes the inverse of the design matrix in the regularization path using the Schur complement. Two cases may arise:

Case 1: one variable is added to the active set

\[\begin{split}M_{k+1}^{-1} =

\begin{bmatrix}

M_{k}^{-1} + s^{-1} M_{k}^{-1} \mathbf{b} \mathbf{b}^T M_{k}^{-1} \

& - M_{k}^{-1} \mathbf{b} s^{-1} \\

- s^{-1} \mathbf{b}^T M_{k}^{-1} & s^{-1}

\end{bmatrix}\end{split}\]</div>

where :

<li>\(M_k^{-1}\) is the inverse of the design matrix \(H_A^tH_A\) of the previous iteration</li>

<li>\(\mathbf{b}\) is the last column of \(M_{k}\)</li>

<li>\(s\) is the Schur complement, given by \(s = \mathbf{d} - \mathbf{b}^T M_{k}^{-1} \mathbf{b}\)</li>

</ul>

Case 2: one variable is removed from the active set.

\[M_{k+1}^{-1} = M^{-1}_{k \backslash q} -

\frac{r_{-q,q} r^{T}_{-q,q}}{r_{q,q}}\]</div>

where :

<li>\(q\) is the index of column and row to delete</li>

<li>\(M^{-1}_{k \backslash q}\) is the previous inverse matrix deprived of the \(q\) th column and \(q\) th row</li>

<li>\(r_{-q,q}\) is the \(q\) th column of \(M^{-1}_{k}\) without the \(q\) th element</li>

<li>\(r_{q, q}\) is the element of \(q\) th column and \(q\) th row in \(M^{-1}_{k}\)</li>

</ul>

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

<li>M_current (ndarray, shape (size(A)-1, size(A)-1)) – Inverse matrix of \(H_A^tH_A\) at the previous iteration, with size(A) the size of the active set</li>

<li>b (ndarray, shape (size(A)-1, )) – None for case 2 (removal), last column of \(M_{k}\) for case 1 (addition)</li>

<li>d (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>) – should be equal to 2 when UOT and 1 for the semi-relaxed OT</li>

<li>id_pop (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>) – Index of the variable to be removed, equal to -1

if no variable is deleted at the current iteration</li>

</ul>

</dd>

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

<dd class="field-even">M – Inverse matrix of \(H_A^tH_A\) of the current iteration

</dd>

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

<dd class="field-odd">ndarray, shape (size(A), size(A))

</dd>

</dl>

References

[41]

Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

</aside>

</dd></dl>

ot.regpath.compute_next_removal(phi, delta, current_gamma)<a class="reference internal" href="../_modules/ot/regpath.html#compute_next_removal">[source]</a><a class="headerlink" href="#ot.regpath.compute_next_removal" title="Link to this definition"></a></dt>

<dd>This function computes the next gamma value if a variable

is removed at the next iteration of the regularization path.

We look for the largest value of the regularization parameter such that

an element of the current solution vanishes

\[\max_{j \in A} \frac{\phi_j}{\delta_j}\]</div>

where :

<li>A is the current active set</li>

<li>\(\phi_j\) is the \(j\) th element of the intercept of the current solution</li>

<li>\(\delta_j\) is the \(j\) th element of the slope of the current solution</li>

</ul>

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

<li>phi (ndarray, shape (size(A), )) – Intercept of the solution at the current iteration</li>

<li>delta (ndarray, shape (size(A), )) – Slope of the solution at the current iteration</li>

<li>current_gamma (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>) – Value of the regularization parameter at the beginning of the current iteration</li>

</ul>

</dd>

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

<li>next_removal_gamma (float) – Gamma value if a variable is removed at the next iteration</li>

<li>next_removal_index (int) – Index of the variable to be removed at the next iteration</li>

</ul>

</dd>

</dl>

References

[41]

Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

</aside>

</dd></dl>

ot.regpath.compute_transport_plan(gamma, gamma_list, Pi_list)<a class="reference internal" href="../_modules/ot/regpath.html#compute_transport_plan">[source]</a><a class="headerlink" href="#ot.regpath.compute_transport_plan" title="Link to this definition"></a></dt>

<dd>Given the regularization path, this function computes the transport

plan for any value of gamma thanks to the piecewise linearity of the path.

\[t(\gamma) = \phi(\gamma) - \gamma \delta(\gamma)\]</div>

where:

<li>\(\gamma\) is the regularization parameter</li>

<li>\(\phi(\gamma)\) is the corresponding intercept</li>

<li>\(\delta(\gamma)\) is the corresponding slope</li>

<li>\(\mathbf{t}\) is the flattened version of the transport matrix</li>

</ul>

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

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

<li>gamma_list (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#list" title="(in Python v3.15)">list</a>) – List of regularization parameters of the regularization path</li>

<li>Pi_list (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#list" title="(in Python v3.15)">list</a>) – List of all the solutions of the regularization path</li>

</ul>

</dd>

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

<dd class="field-even">t – Vectorization of the transport plan corresponding to the given value

of gamma

</dd>

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

<dd class="field-odd">np.ndarray (dim_a*dim_b, )

</dd>

</dl>

Examples

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

>>> import numpy as np

>>> n = 3

>>> xs = np.array([1., 2., 3.]).reshape((n, 1))

>>> xt = np.array([5., 6., 7.]).reshape((n, 1))

>>> C = ot.dist(xs, xt)

>>> C /= C.max()

>>> a = np.array([0.2, 0.5, 0.3])

>>> b = np.array([0.2, 0.5, 0.3])

>>> t, pi_list, g_list = ot.regpath.regularization_path(a, b, C, reg=1e-4)

>>> gamma = 1

>>> t2 = ot.regpath.compute_transport_plan(gamma, g_list, pi_list)

>>> t2

array([0. , 0. , 0. , 0.19722222, 0.05555556,

0. , 0. , 0.24722222, 0. ])

</pre></div>

</div>

References

[41]

Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

</aside>

</dd></dl>

<h3>Examples using <code class="docutils literal notranslate">ot.regpath.compute_transport_plan</code><a class="headerlink" href="#examples-using-ot-regpath-compute-transport-plan" title="Link to this heading"></a></h3>

<a class="reference internal" href="../auto_examples/unbalanced-partial/plot_regpath.html">Regularization path of l2-penalized unbalanced optimal transport</a>

<div class="sphx-glr-thumbnail-title">Regularization path of l2-penalized unbalanced optimal transport</div>

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

ot.regpath.construct_augmented_H(active_index, m, Hc, HrHr)<a class="reference internal" href="../_modules/ot/regpath.html#construct_augmented_H">[source]</a><a class="headerlink" href="#ot.regpath.construct_augmented_H" title="Link to this definition"></a></dt>

<dd>This function constructs an augmented matrix for the first iteration

of the semi-relaxed regularization path

\[\begin{split}\text{Augmented}_H =

\begin{bmatrix}

0 & H_{c A} \\

H_{c A}^T & H_{r A}^T H_{r A}

\end{bmatrix}\end{split}\]</div>

where :

<li>\(H_{r A}\) is the sub-matrix constructed by the columns of \(H_r\) whose indices belong to the active set A</li>

<li>\(H_{c A}\) is the sub-matrix constructed by the columns of \(H_c\) whose indices belong to the active set A</li>

</ul>

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

<li>active_index (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#list" title="(in Python v3.15)">list</a>) – Indices of the active variables</li>

<li>m (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>) – Length of the target distribution</li>

<li>Hc (np.ndarray (dim_b, dim_a * dim_b)) – Matrix that computes the sum along the columns of the transport plan \(T\)</li>

<li>HrHr (np.ndarray (dim_a * dim_b, dim_a * dim_b)) – Matrix product of \(H_r^T H_r\)</li>

</ul>

</dd>

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

<dd class="field-even">H_augmented – Augmented matrix for the first iteration of the semi-relaxed

regularization path

</dd>

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

<dd class="field-odd">np.ndarray (dim_b + size(A), dim_b + size(A))

</dd>

</dl>

</dd></dl>

ot.regpath.fully_relaxed_path(a: array, b: array, C: array, reg=0.0001, itmax=50000)<a class="reference internal" href="../_modules/ot/regpath.html#fully_relaxed_path">[source]</a><a class="headerlink" href="#ot.regpath.fully_relaxed_path" title="Link to this definition"></a></dt>

<dd>This function gives the regularization path of l2-penalized UOT problem

The problem to optimize is the Lasso reformulation of the l2-penalized UOT:

\[ \begin{align}\begin{aligned}\min_t \gamma \mathbf{c}^T \mathbf{t}

+ 0.5 * \|{H} \mathbf{t} - \mathbf{y}\|_2^2\\s.t.

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

where :

<li>\(\mathbf{c}\) is the flattened version of the cost matrix \({C}\)</li>

<li>\(\gamma = 1/\lambda\) is the l2-regularization coefficient</li>

<li>\(\mathbf{y}\) is the concatenation of vectors \(\mathbf{a}\) and \(\mathbf{b}\), defined as \(\mathbf{y}^T = [\mathbf{a}^T \mathbf{b}^T]\)</li>

<li>\({H}\) is a design matrix, see <a class="reference internal" href="ot.unbalanced.html#references-regpath">[41]</a> for the design of \({H}\). The matrix product \(H\mathbf{t}\) computes both the source marginal and the target marginals.</li>

<li>\(\mathbf{t}\) is the flattened version of the transport matrix</li>

</ul>

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

<li>a (np.ndarray (dim_a,)) – Histogram of dimension dim_a</li>

<li>b (np.ndarray (dim_b,)) – Histogram of dimension dim_b</li>

<li>C (np.ndarray, shape (dim_a, dim_b)) – Cost matrix</li>

<li>reg (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>) – l2-regularization coefficient</li>

<li>itmax (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>) – Maximum number of iteration</li>

</ul>

</dd>

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

<li>t (np.ndarray (dim_a*dim_b, )) – Flattened vector of the optimal transport matrix</li>

<li>t_list (list) – List of solutions in the regularization path</li>

<li>gamma_list (list) – List of regularization coefficients in the regularization path</li>

</ul>

</dd>

</dl>

Examples

>>> n = 3

>>> C /= C.max()

>>> t, _, _ = ot.regpath.fully_relaxed_path(a, b, C, 1e-4)

>>> t

array([1.99958333e-01, 0.00000000e+00, 0.00000000e+00, 3.88888889e-05,

4.99938889e-01, 0.00000000e+00, 0.00000000e+00, 3.88888889e-05,

2.99958333e-01])

</pre></div>

</div>

References

[41]

Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

</aside>

</dd></dl>

ot.regpath.ot_next_gamma(phi, delta, HtH, Hty, c, active_index, current_gamma)<a class="reference internal" href="../_modules/ot/regpath.html#ot_next_gamma">[source]</a><a class="headerlink" href="#ot.regpath.ot_next_gamma" title="Link to this definition"></a></dt>

<dd>This function computes the next value of gamma if a variable

is added in the next iteration of the regularization path.

We look for the largest value of gamma such that

the gradient of an inactive variable vanishes

\[\max_{i \in \bar{A}} \frac{\mathbf{h}_i^T(H_A \phi - \mathbf{y})}

{\mathbf{h}_i^T H_A \delta - \mathbf{c}_i}\]</div>

where :

<li>A is the current active set</li>

<li>\(\mathbf{h}_i\) is the \(i\) th column of the design matrix \({H}\)</li>

<li>\({H}_A\) is the sub-matrix constructed by the columns of \({H}\) whose indices belong to the active set A</li>

<li>\(\mathbf{c}_i\) is the \(i\) th element of the cost vector \(\mathbf{c}\)</li>

<li>\(\mathbf{y}\) is the concatenation of the source and target distributions</li>

<li>\(\phi\) is the intercept of the solutions at the current iteration</li>

<li>\(\delta\) is the slope of the solutions at the current iteration</li>

</ul>

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

<li>phi (np.ndarray (size(A), )) – Intercept of the solutions at the current iteration</li>

<li>delta (np.ndarray (size(A), )) – Slope of the solutions at the current iteration</li>

<li>HtH (np.ndarray (dim_a * dim_b, dim_a * dim_b)) – Matrix product of \({H}^T {H}\)</li>

<li>Hty (np.ndarray (dim_a + dim_b, )) – Matrix product of \({H}^T \mathbf{y}\)</li>

<li>c (np.ndarray (dim_a * dim_b, )) – Flattened array of the cost matrix \({C}\)</li>

</ul>

</dd>

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

<li>next_gamma (float) – Value of gamma if a variable is added to active set in next iteration</li>

<li>next_active_index (int) – Index of variable to be activated</li>

</ul>

</dd>

</dl>

References

[41]

Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

</aside>

</dd></dl>

ot.regpath.recast_ot_as_lasso(a, b, C)<a class="reference internal" href="../_modules/ot/regpath.html#recast_ot_as_lasso">[source]</a><a class="headerlink" href="#ot.regpath.recast_ot_as_lasso" title="Link to this definition"></a></dt>

<dd>This function recasts the l2-penalized UOT problem as a Lasso problem.

Recall the l2-penalized UOT problem defined in

\[ \begin{align}\begin{aligned}\text{UOT}_{\lambda} = \min_T <C, T> + \lambda \|T 1_m -

\mathbf{a}\|_2^2 +

\lambda \|T^T 1_n - \mathbf{b}\|_2^2\\s.t.

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

where :

<li>\(C\) is the cost matrix</li>

<li>\(\lambda\) is the l2-regularization parameter</li>

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

<li>\(T\) is the transport plan to optimize</li>

</ul>

The problem above can be reformulated as a non-negative penalized

linear regression problem, particularly Lasso

\[ \begin{align}\begin{aligned}\text{UOT2}_{\lambda} = \min_{\mathbf{t}} \gamma \mathbf{c}^T

\mathbf{t} + 0.5 * \|H \mathbf{t} - \mathbf{y}\|_2^2\\s.t.

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

where :

<li>\(\mathbf{c}\) is the flattened version of the cost matrix \(C\)</li>

<li>\(H\) is a metric matrix, see <a class="reference internal" href="ot.unbalanced.html#references-regpath">[41]</a> for the design of \(H\). The matrix product \(H\mathbf{t}\) computes both the source marginal and the target marginals.</li>

<li>\(\mathbf{t}\) is the flattened version of the transport plan \(T\)</li>

</ul>

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

<li>a (np.ndarray (dim_a,)) – Histogram of dimension dim_a</li>

<li>b (np.ndarray (dim_b,)) – Histogram of dimension dim_b</li>

<li>C (np.ndarray, shape (dim_a, dim_b)) – Cost matrix</li>

</ul>

</dd>

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

<li>H (np.ndarray (dim_a+dim_b, dim_a*dim_b)) – Design matrix that contains only 0 and 1</li>

<li>y (np.ndarray (ns + nt, )) – Concatenation of histograms \(\mathbf{a}\) and \(\mathbf{b}\)</li>

<li>c (np.ndarray (ns * nt, )) – Flattened array of the cost matrix</li>

</ul>

</dd>

</dl>

Examples

>>> a = np.array([0.2, 0.3, 0.5])

>>> b = np.array([0.1, 0.9])

>>> C = np.array([[16., 25.], [28., 16.], [40., 36.]])

>>> H, y, c = ot.regpath.recast_ot_as_lasso(a, b, C)

>>> H.toarray()

array([[1., 1., 0., 0., 0., 0.],

[0., 0., 1., 1., 0., 0.],

[0., 0., 0., 0., 1., 1.],

[1., 0., 1., 0., 1., 0.],

[0., 1., 0., 1., 0., 1.]])

>>> y

array([0.2, 0.3, 0.5, 0.1, 0.9])

>>> c

array([16., 25., 28., 16., 40., 36.])

</pre></div>

</div>

References

[41]

Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

</aside>

</dd></dl>

ot.regpath.recast_semi_relaxed_as_lasso(a, b, C)<a class="reference internal" href="../_modules/ot/regpath.html#recast_semi_relaxed_as_lasso">[source]</a><a class="headerlink" href="#ot.regpath.recast_semi_relaxed_as_lasso" title="Link to this definition"></a></dt>

<dd>This function recasts the semi-relaxed l2-UOT problem as Lasso problem.

\[ \begin{align}\begin{aligned}\text{semi-relaxed UOT} = \min_T <C, T>

+ \lambda \|T 1_m - \mathbf{a}\|_2^2\\s.t.

T^T 1_n = \mathbf{b}\\ \mathbf{t} \geq 0\end{aligned}\end{align} \]</div>

where :

<li>\(C\) is the metric cost matrix</li>

<li>\(\lambda\) is the l2-regularization parameter</li>

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

<li>\(T\) is the transport plan to optimize</li>

</ul>

The problem above can be reformulated as follows

\[ \begin{align}\begin{aligned}\text{semi-relaxed UOT2} = \min_t \gamma \mathbf{c}^T t

+ 0.5 * \|H_r \mathbf{t} - \mathbf{a}\|_2^2\\s.t.

H_c \mathbf{t} = \mathbf{b}\\ \mathbf{t} \geq 0\end{aligned}\end{align} \]</div>

where :

<li>\(\mathbf{c}\) is flattened version of the cost matrix \(C\)</li>

<li>\(\gamma = 1/\lambda\) is the l2-regularization parameter</li>

<li>\(H_r\) is a metric matrix which computes the sum along the rows of the transport plan \(T\)</li>

<li>\(H_c\) is a metric matrix which computes the sum along the columns of the transport plan \(T\)</li>

<li>\(\mathbf{t}\) is the flattened version of \(T\)</li>

</ul>

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

<li>a (np.ndarray (dim_a,)) – Histogram of dimension dim_a</li>

<li>b (np.ndarray (dim_b,)) – Histogram of dimension dim_b</li>

<li>C (np.ndarray, shape (dim_a, dim_b)) – Cost matrix</li>

</ul>

</dd>

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

<li>Hr (np.ndarray (dim_a, dim_a * dim_b)) – Auxiliary matrix constituted by 0 and 1, which computes

the sum along the rows of transport plan \(T\)</li>

<li>Hc (np.ndarray (dim_b, dim_a * dim_b)) – Auxiliary matrix constituted by 0 and 1, which computes

the sum along the columns of transport plan \(T\)</li>

<li>c (np.ndarray (ns * nt, )) – Flattened array of the cost matrix</li>

</ul>

</dd>

</dl>

Examples

>>> a = np.array([0.2, 0.3, 0.5])

>>> b = np.array([0.1, 0.9])

>>> Hr,Hc,c = ot.regpath.recast_semi_relaxed_as_lasso(a, b, C)

>>> Hr.toarray()

array([[1., 1., 0., 0., 0., 0.],

[0., 0., 1., 1., 0., 0.],

[0., 0., 0., 0., 1., 1.]])

>>> Hc.toarray()

array([[1., 0., 1., 0., 1., 0.],

[0., 1., 0., 1., 0., 1.]])

>>> c

array([16., 25., 28., 16., 40., 36.])

</pre></div>

</div>

</dd></dl>

ot.regpath.regularization_path(a: array, b: array, C: array, reg=0.0001, semi_relaxed=False, itmax=50000)<a class="reference internal" href="../_modules/ot/regpath.html#regularization_path">[source]</a><a class="headerlink" href="#ot.regpath.regularization_path" title="Link to this definition"></a></dt>

<dd>This function provides all the solutions of the regularization path of the l2-UOT problem <a class="reference internal" href="ot.unbalanced.html#references-regpath">[41]</a>.

The problem to optimize is the Lasso reformulation of the l2-penalized UOT:

\[ \begin{align}\begin{aligned}\min_t \gamma \mathbf{c}^T \mathbf{t}

+ 0.5 * \|{H} \mathbf{t} - \mathbf{y}\|_2^2\\s.t.

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

where :

<li>\(\mathbf{c}\) is the flattened version of the cost matrix \({C}\)</li>

<li>\(\gamma = 1/\lambda\) is the l2-regularization coefficient</li>

<li>\(\mathbf{t}\) is the flattened version of the transport matrix</li>

</ul>

For the semi-relaxed problem, it optimizes the Lasso reformulation of the

l2-penalized UOT:

\[ \begin{align}\begin{aligned}\min_t \gamma \mathbf{c}^T \mathbf{t}

+ 0.5 * \|H_r \mathbf{t} - \mathbf{a}\|_2^2\\s.t.

H_c \mathbf{t} = \mathbf{b}\\ \mathbf{t} \geq 0\end{aligned}\end{align} \]</div>

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

<li>a (np.ndarray (dim_a,)) – Histogram of dimension dim_a</li>

<li>b (np.ndarray (dim_b,)) – Histogram of dimension dim_b</li>

<li>C (np.ndarray, shape (dim_a, dim_b)) – Cost matrix</li>

<li>semi_relaxed (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#bool" title="(in Python v3.15)">bool</a> (optional)) – Give the semi-relaxed path if True</li>

</ul>

</dd>

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

<li>t (np.ndarray (dim_a*dim_b, )) – Flattened vector of the (unregularized) optimal transport matrix</li>

<li>t_list (list) – List of all the optimal transport vectors of the regularization path</li>

<li>gamma_list (list) – List of the regularization parameters in the path</li>

</ul>

</dd>

</dl>

References

[41]

Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

</aside>

</dd></dl>

</section>

<h3>Examples using <code class="docutils literal notranslate">ot.regpath.regularization_path</code><a class="headerlink" href="#examples-using-ot-regpath-regularization-path" title="Link to this heading"></a></h3>

<a class="reference internal" href="../auto_examples/unbalanced-partial/plot_regpath.html">Regularization path of l2-penalized unbalanced optimal transport</a>

<div class="sphx-glr-thumbnail-title">Regularization path of l2-penalized unbalanced optimal transport</div>

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

ot.regpath.semi_relaxed_next_gamma(phi, delta, phi_u, delta_u, HrHr, Hc, Hra, c, active_index, current_gamma)<a class="reference internal" href="../_modules/ot/regpath.html#semi_relaxed_next_gamma">[source]</a><a class="headerlink" href="#ot.regpath.semi_relaxed_next_gamma" title="Link to this definition"></a></dt>

<dd>This function computes the next value of gamma when a variable is

active in the regularization path of semi-relaxed UOT.

By taking the Lagrangian form of the problem, we obtain a similar update

as the two-sided relaxed UOT

\[\max_{i \in \bar{A}} \frac{\mathbf{h}_{ri}^T(H_{rA} \phi - \mathbf{a})

+ \mathbf{h}_{c i}^T\phi_u}{\mathbf{h}_{r i}^T H_{r A} \delta + \

\mathbf{h}_{c i} \delta_u - \mathbf{c}_i}\]</div>

where :

<li>A is the current active set</li>

<li>\(\mathbf{h}_{r i}\) is the ith column of the matrix \(H_r\)</li>

<li>\(\mathbf{h}_{c i}\) is the ith column of the matrix \(H_c\)</li>

<li>\(\mathbf{c}_i\) is the \(i\) th element of cost vector \(\mathbf{c}\)</li>

<li>\(\phi\) is the intercept of the solutions in current iteration</li>

<li>\(\delta\) is the slope of the solutions in current iteration</li>

<li>\(\phi_u\) is the intercept of Lagrange parameter at the current iteration</li>

<li>\(\delta_u\) is the slope of Lagrange parameter at the current iteration</li>

</ul>

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

<li>phi (np.ndarray (size(A), )) – Intercept of the solutions at the current iteration</li>

<li>delta (np.ndarray (size(A), )) – Slope of the solutions at the current iteration</li>

<li>phi_u (np.ndarray (dim_b, )) – Intercept of the Lagrange parameter at the current iteration</li>

<li>delta_u (np.ndarray (dim_b, )) – Slope of the Lagrange parameter at the current iteration</li>

<li>Hra (np.ndarray (dim_a * dim_b, )) – Matrix product of \(H_r^T \mathbf{a}\)</li>

<li>c (np.ndarray (dim_a * dim_b, )) – Flattened array of cost matrix \(C\)</li>

<li>current_gamma (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>) – Value of regularization coefficient at the start of current iteration</li>

</ul>

</dd>

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

<li>next_gamma (float) – Value of gamma if a variable is added to active set in next iteration</li>

<li>next_active_index (int) – Index of variable to be activated</li>

</ul>

</dd>

</dl>

References

[41]

Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

</aside>

</dd></dl>

ot.regpath.semi_relaxed_path(a: array, b: array, C: array, reg=0.0001, itmax=50000)<a class="reference internal" href="../_modules/ot/regpath.html#semi_relaxed_path">[source]</a><a class="headerlink" href="#ot.regpath.semi_relaxed_path" title="Link to this definition"></a></dt>

<dd>This function gives the regularization path of semi-relaxed

l2-UOT problem.

The problem to optimize is the Lasso reformulation of the l2-penalized UOT:

\[ \begin{align}\begin{aligned}\min_t \gamma \mathbf{c}^T t

+ 0.5 * \|H_r \mathbf{t} - \mathbf{a}\|_2^2\\s.t.

H_c \mathbf{t} = \mathbf{b}\\ \mathbf{t} \geq 0\end{aligned}\end{align} \]</div>

where :

<li>\(\mathbf{c}\) is the flattened version of the cost matrix \(C\)</li>

<li>\(\gamma = 1/\lambda\) is the l2-regularization parameter</li>

<li>\(H_r\) is a matrix that computes the sum along the rows of the transport plan \(T\)</li>

<li>\(H_c\) is a matrix that computes the sum along the columns of the transport plan \(T\)</li>

<li>\(\mathbf{t}\) is the flattened version of the transport plan \(T\)</li>

</ul>

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

<li>a (np.ndarray (dim_a,)) – Histogram of dimension dim_a</li>

<li>b (np.ndarray (dim_b,)) – Histogram of dimension dim_b</li>

<li>C (np.ndarray, shape (dim_a, dim_b)) – Cost matrix</li>

</ul>

</dd>

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

<li>t (np.ndarray (dim_a*dim_b, )) – Flattened vector of the (unregularized) optimal transport matrix</li>

<li>t_list (list) – List of all the optimal transport vectors of the regularization path</li>

<li>gamma_list (list) – List of the regularization parameters in the path</li>

</ul>

</dd>

</dl>

Examples

>>> n = 3

>>> C /= C.max()

>>> t, _, _ = ot.regpath.semi_relaxed_path(a, b, C, 1e-4)

>>> t

array([1.99980556e-01, 0.00000000e+00, 0.00000000e+00, 1.94444444e-05,

4.99980556e-01, 0.00000000e+00, 0.00000000e+00, 1.94444444e-05,

3.00000000e-01])

</pre></div>

</div>

References

[41]

Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

</aside>

</dd></dl>

</section>

<dd>This function computes the inverse of the design matrix in the regularization path using the Schur complement. Two cases may arise:

Case 1: one variable is added to the active set

\[\begin{split}M_{k+1}^{-1} =

\begin{bmatrix}

M_{k}^{-1} + s^{-1} M_{k}^{-1} \mathbf{b} \mathbf{b}^T M_{k}^{-1} \

& - M_{k}^{-1} \mathbf{b} s^{-1} \\

- s^{-1} \mathbf{b}^T M_{k}^{-1} & s^{-1}

\end{bmatrix}\end{split}\]</div>

where :

<li>\(\mathbf{b}\) is the last column of \(M_{k}\)</li>

</ul>

Case 2: one variable is removed from the active set.

\[M_{k+1}^{-1} = M^{-1}_{k \backslash q} -

\frac{r_{-q,q} r^{T}_{-q,q}}{r_{q,q}}\]</div>

where :

<li>\(q\) is the index of column and row to delete</li>

</ul>

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

if no variable is deleted at the current iteration</li>

</ul>

</dd>

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

<dd class="field-even">M – Inverse matrix of \(H_A^tH_A\) of the current iteration

</dd>

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

<dd class="field-odd">ndarray, shape (size(A), size(A))

</dd>

</dl>

References

[41]

Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

</aside>

</dd></dl>

<dd>This function computes the next gamma value if a variable

is removed at the next iteration of the regularization path.

We look for the largest value of the regularization parameter such that

an element of the current solution vanishes

\[\max_{j \in A} \frac{\phi_j}{\delta_j}\]</div>

where :

<li>A is the current active set</li>

<li>\(\phi_j\) is the \(j\) th element of the intercept of the current solution</li>

<li>\(\delta_j\) is the \(j\) th element of the slope of the current solution</li>

</ul>

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

</ul>

</dd>

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

<li>next_removal_gamma (float) – Gamma value if a variable is removed at the next iteration</li>

<li>next_removal_index (int) – Index of the variable to be removed at the next iteration</li>

</ul>

</dd>

</dl>

References

[41]

Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

</aside>

</dd></dl>

<dd>Given the regularization path, this function computes the transport

plan for any value of gamma thanks to the piecewise linearity of the path.

\[t(\gamma) = \phi(\gamma) - \gamma \delta(\gamma)\]</div>

where:

<li>\(\gamma\) is the regularization parameter</li>

<li>\(\phi(\gamma)\) is the corresponding intercept</li>

<li>\(\delta(\gamma)\) is the corresponding slope</li>

<li>\(\mathbf{t}\) is the flattened version of the transport matrix</li>

</ul>

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

</ul>

</dd>

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

<dd class="field-even">t – Vectorization of the transport plan corresponding to the given value

of gamma

</dd>

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

<dd class="field-odd">np.ndarray (dim_a*dim_b, )

</dd>

</dl>

Examples

>>> n = 3

>>> C /= C.max()

>>> gamma = 1

>>> t2

array([0. , 0. , 0. , 0.19722222, 0.05555556,

0. , 0. , 0.24722222, 0. ])

</pre></div>

</div>

References

[41]

Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

</aside>

</dd></dl>

<dd>This function constructs an augmented matrix for the first iteration

of the semi-relaxed regularization path

\[\begin{split}\text{Augmented}_H =

\begin{bmatrix}

0 & H_{c A} \\

H_{c A}^T & H_{r A}^T H_{r A}

\end{bmatrix}\end{split}\]</div>

where :

</ul>

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

</ul>

</dd>

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

<dd class="field-even">H_augmented – Augmented matrix for the first iteration of the semi-relaxed

regularization path

</dd>

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

<dd class="field-odd">np.ndarray (dim_b + size(A), dim_b + size(A))

</dd>

</dl>

</dd></dl>

<dd>This function gives the regularization path of l2-penalized UOT problem

The problem to optimize is the Lasso reformulation of the l2-penalized UOT:

\[ \begin{align}\begin{aligned}\min_t \gamma \mathbf{c}^T \mathbf{t}

+ 0.5 * \|{H} \mathbf{t} - \mathbf{y}\|_2^2\\s.t.

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

where :

<li>\(\mathbf{c}\) is the flattened version of the cost matrix \({C}\)</li>

<li>\(\gamma = 1/\lambda\) is the l2-regularization coefficient</li>

<li>\(\mathbf{t}\) is the flattened version of the transport matrix</li>

</ul>

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

<li>a (np.ndarray (dim_a,)) – Histogram of dimension dim_a</li>

<li>b (np.ndarray (dim_b,)) – Histogram of dimension dim_b</li>

<li>C (np.ndarray, shape (dim_a, dim_b)) – Cost matrix</li>

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