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<!DOCTYPE html>

<head>

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

</head>

POT Python Optimal Transport

</a>

</form>

</div>

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

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

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

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

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

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

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

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

<li class="toctree-l3"><a class="reference internal" href="#ot.lp.dmmot_monge_1dgrid_loss"><code class="docutils literal notranslate">dmmot_monge_1dgrid_loss()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#ot.lp.dmmot_monge_1dgrid_optimize"><code class="docutils literal notranslate">dmmot_monge_1dgrid_optimize()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#ot.lp.emd_1d_dual_backprop"><code class="docutils literal notranslate">emd_1d_dual_backprop()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#ot.lp.emd_1d_sorted"><code class="docutils literal notranslate">emd_1d_sorted()</code></a></li>

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

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

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

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

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

<li class="toctree-l3"><a class="reference internal" href="#ot.lp.semidiscrete_wasserstein2_unif_circle"><code class="docutils literal notranslate">semidiscrete_wasserstein2_unif_circle()</code></a></li>

<li class="toctree-l3"><a class="reference internal" href="#ot.lp.wasserstein_1d"><code class="docutils literal notranslate">wasserstein_1d()</code></a></li>

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

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

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

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

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

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

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

<li class="toctree-l2"><a class="reference internal" href="../all.html#module-ot">Main <code class="xref py py-mod docutils literal notranslate">ot</code> functions</a></li>

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

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

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<li class="breadcrumb-item"><a href="../all.html">API and modules</a></li>

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

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<hr/>

</div>

Solvers for the original linear program OT problem.

ot.lp.NorthWestMMGluing(pi_list, a=None, log=False, nx=None)<a class="reference internal" href="../_modules/ot/lp/_barycenter_solvers.html#NorthWestMMGluing">[source]</a><a class="headerlink" href="#ot.lp.NorthWestMMGluing" title="Link to this definition"></a></dt>

<dd>Glue transport plans $(\pi_1, ..., \pi_K)$ which have a common first

marginal using the (multi-marginal) North-West Corner method. Writing the

marginals of each $\pi_k\in \mathbb{R}^{n\times n_l}$ as \(a \in

\mathbb{R}^n\) and $b_k \in \mathbb{R}^{n_k}$, the output represents a

particular K-marginal transport plan \(\rho \in

\mathbb{R}^{n_1\times\cdots\times n_K}\) whose k-th marginal is $b_k$.

This K-plan is such that there exists a K+1-marginal transport plan

$\gamma \in \mathbb{R}^{n\times n_1 \times \cdots \times n_K}$ such

that $\sum_i\gamma_{i,j_1,\cdots,j_K} = \rho_{j_1, \cdots, j_K}$ and

with Einstein summation convention, \(\gamma_{i, j_1, \cdots, j_K} =

[\pi_k]_{i, j_k}\) for all $k=1,\cdots,K$.

Instead of outputting the full K-multi-marginal plan $\rho$, this

function provides an array <cite>J</cite> of shape (N, K) where each <cite>J[i]</cite> is of the

form <cite>(J[i, 1], …, J[i, K])</cite> with each <cite>J[i, k]</cite> between 0 and

$n_k-1$, and a weight vector <cite>w</cite> of size N, such that the K-plan

$\rho$ writes:

\[\rho_{j_1, \cdots, j_K} = \mathbb{1}\left(\exists i \text{ s.t. } (j_1, \cdots, j_K) = (J[i, 1], \cdots, J[i, K])\right)\ w_i.\]</div>

This representation is useful for its memory efficiency, as it avoids

storing the full K-marginal plan.

If <cite>log=True</cite>, the function computes the full K+1-marginal transport plan

$\gamma$ and stores it in log_dict[‘gamma’]. Note that this option is

extremely costly in memory.

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

<li>pi_list (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#list" title="(in Python v3.15)">list</a> of arrays (n, n_k)) – List of transport plans.</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) – If True, return a log dictionary (computationally expensive).</li>

<li>a (array (n,), optional) – The common first marginal of each transport plan. If None is provided,

it is computed as the first marginal of the first transport plan.</li>

<li>nx (<a class="reference internal" href="ot.utils.html#id32" title="ot.utils.BaryResult.backend">backend</a>, optional) – The backend to use. If None is provided, the backend of the first

transport plan is used.</li>

</ul>

</dd>

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

<li>J (array (N, K)) – The indices (J[i, 1], …, J[i, K]) of the K-plan $\rho$.</li>

<li>w (array (N,)) – The weights w_i of the K-plan $\rho$.</li>

<li>log_dict (dict, optional) – If log=True, a dictionary containing the full K+1-marginal transport

plan under the key ‘gamma’.</li>

</ul>

</dd>

</dl>

</dd></dl>

ot.lp.barycenter(A, M, weights=None, verbose=False, log=False, solver='highs-ipm')<a class="reference internal" href="../_modules/ot/lp/_barycenter_solvers.html#barycenter">[source]</a><a class="headerlink" href="#ot.lp.barycenter" title="Link to this definition"></a></dt>

<dd>Compute the Wasserstein barycenter of distributions A

<div>The function solves the following optimization problem [16]:

</div></blockquote>

\[\mathbf{a} = arg\min_\mathbf{a} \sum_i W_{1}(\mathbf{a},\mathbf{a}_i)\]</div>

where :

<li>$W_1(\cdot,\cdot)$ is the Wasserstein distance (see ot.emd.sinkhorn)</li>

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

</ul>

The linear program is solved using the interior point solver from scipy.optimize.

If cvxopt solver if installed it can use cvxopt

Note that this problem do not scale well (both in memory and computational time).

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

<li>A (np.ndarray (d,n)) – n training distributions a_i of size d</li>

<li>M (np.ndarray (d,d)) – loss matrix for OT</li>

<li>weights (np.ndarray (n,)) – Weights of each histogram a_i on the simplex (barycentric coordinates)</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>solver (string, optional) – the solver used, default ‘interior-point’ use the lp solver from

scipy.optimize. None, or ‘glpk’ or ‘mosek’ use the solver from cvxopt.</li>

</ul>

</dd>

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

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

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

</ul>

</dd>

</dl>

References

[16]

Agueh, M., & Carlier, G. (2011). Barycenters in the Wasserstein space. SIAM Journal on Mathematical Analysis, 43(2), 904-924.

</aside>

</dd></dl>

ot.lp.binary_search_circle(u_values, v_values, u_weights=None, v_weights=None, p=1, Lm=10, Lp=10, tm=-1, tp=1, eps=1e-06, require_sort=True, log=False)<a class="reference internal" href="../_modules/ot/lp/solver_circle.html#binary_search_circle">[source]</a><a class="headerlink" href="#ot.lp.binary_search_circle" title="Link to this definition"></a></dt>

<dd>Computes the Wasserstein distance on the circle using the Binary search algorithm proposed in [44].

Samples need to be in $S^1\cong [0,1[$. If they are on $\mathbb{R}$,

takes the value modulo 1.

If the values are on $S^1\subset\mathbb{R}^2$, it is required to first find the coordinates

using e.g. the atan2 function.

\[W_p^p(u,v) = \inf_{\theta\in\mathbb{R}}\int_0^1 |F_u^{-1}(q) - (F_v-\theta)^{-1}(q)|^p\ \mathrm{d}q\]</div>

where:

<li>$F_u$ and $F_v$ are respectively the cdfs of $u$ and $v$</li>

</ul>

For values $x=(x_1,x_2)\in S^1$, it is required to first get their coordinates with

\[u = \frac{\pi + \mathrm{atan2}(-x_2,-x_1)}{2\pi}\]</div>

using e.g. ot.utils.get_coordinate_circle(x)

The function runs on backend but tensorflow and jax are not supported.

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

<li>u_values (ndarray, shape (n, ...)) – samples in the source domain (coordinates on [0,1[)</li>

<li>v_values (ndarray, shape (n, ...)) – samples in the target domain (coordinates on [0,1[)</li>

<li>u_weights (ndarray, shape (n, ...), optional) – samples weights in the source domain</li>

<li>v_weights (ndarray, shape (n, ...), optional) – samples weights in the target domain</li>

<li>p (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>, optional (default=1)) – Power p used for computing the Wasserstein distance</li>

<li>Lm (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional) – Lower bound dC</li>

<li>Lp (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional) – Upper bound dC</li>

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

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

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

<li>require_sort (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#bool" title="(in Python v3.15)">bool</a>, optional) – If True, sort the values.</li>

</ul>

</dd>

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

<li>loss (float/array-like, shape (…)) – Batched cost associated to the optimal transportation</li>

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

</ul>

</dd>

</dl>

Examples

<div class="doctest highlight-default notranslate"><div class="highlight"><pre>>>> u = np.array([[0.2,0.5,0.8]])%1

>>> v = np.array([[0.4,0.5,0.7]])%1

>>> binary_search_circle(u.T, v.T, p=1)

array([0.1])

</pre></div>

</div>

References

[44]

Delon, Julie, Julien Salomon, and Andrei Sobolevski. “Fast transport optimization for Monge costs on the circle.” SIAM Journal on Applied Mathematics 70.7 (2010): 2239-2258.

</aside>

</dd></dl>

ot.lp.check_number_threads(numThreads)<a class="reference internal" href="../_modules/ot/utils.html#check_number_threads">[source]</a><a class="headerlink" href="#ot.lp.check_number_threads" title="Link to this definition"></a></dt>

<dd>Checks whether or not the requested number of threads has a valid value.

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

<dd class="field-odd">numThreads (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a> or <a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)">str</a>) – The requested number of threads, should either be a strictly positive integer or “max” or None

</dd>

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

<dd class="field-even">numThreads – Corrected number of threads

</dd>

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

</dd>

</dl>

</dd></dl>

ot.lp.dmmot_monge_1dgrid_loss(A, verbose=False, log=False)<a class="reference internal" href="../_modules/ot/lp/dmmot.html#dmmot_monge_1dgrid_loss">[source]</a><a class="headerlink" href="#ot.lp.dmmot_monge_1dgrid_loss" title="Link to this definition"></a></dt>

<dd>Compute the discrete multi-marginal optimal transport of distributions A.

This function operates on distributions whose supports are real numbers on

the real line.

The algorithm solves both primal and dual d-MMOT programs concurrently to

produce the optimal transport plan as well as the total (minimal) cost.

The cost is a ground cost, and the solution is independent of

which Monge cost is desired.

The algorithm accepts $d$ distributions (i.e., histograms)

$a_{1}, \ldots, a_{d} \in \mathbb{R}_{+}^{n}$ with \(e^{\prime}

a_{j}=1\) for all $j \in[d]$. Although the algorithm states that all

histograms have the same number of bins, the algorithm can be easily

adapted to accept as inputs $a_{i} \in \mathbb{R}_{+}^{n_{i}}$

with $n_{i} \neq n_{j}$ [50].

The function solves the following optimization problem[51]:

\[\begin{split}\begin{align}\begin{aligned}

\underset{\gamma\in\mathbb{R}^{n^{d}}_{+}} {\textrm{min}}

\sum_{i_1,\ldots,i_d} c(i_1,\ldots, i_d)\, \gamma(i_1,\ldots,i_d)

\quad \textrm{s.t.}

\sum_{i_2,\ldots,i_d} \gamma(i_1,\ldots,i_d) &= a_1(i_i),

(\forall i_1\in[n])\\

\qquad\vdots\\

\sum_{i_1,\ldots,i_{d-1}} \gamma(i_1,\ldots,i_d) &= a_{d}(i_{d}),

(\forall i_d\in[n]).

\end{aligned}

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

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

<li>A (nx.ndarray, shape (dim, n_hists)) – The input ndarray containing distributions of n bins in d dimensions.</li>

</ul>

</dd>

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

<li>obj (float) – the value of the primal objective function evaluated at the solution.</li>

<li>log (dict) – A dictionary containing the log of the discrete mmot problem:

- ‘A’: a dictionary that maps tuples of indices to the corresponding

primal variables. The tuples are the indices of the entries that are

set to their minimum value during the algorithm.

- ‘primal objective’: a float, the value of the objective function

evaluated at the solution.

- ‘dual’: a list of arrays, the dual variables corresponding to

the input arrays. The i-th element of the list is the dual variable

corresponding to the i-th dimension of the input arrays.

- ‘dual objective’: a float, the value of the dual objective function

evaluated at the solution.</li>

</ul>

</dd>

</dl>

References

[55]

Ronak Mehta, Jeffery Kline, Vishnu Suresh Lokhande, Glenn Fung, &

Vikas Singh (2023). Efficient Discrete Multi Marginal Optimal

Transport Regularization. In The Eleventh International

Conference on Learning Representations.

</aside>

[56]

Jeffery Kline. Properties of the d-dimensional earth mover’s

problem. Discrete Applied Mathematics, 265: 128-141, 2019.

</aside>

[58]

Leonid V Kantorovich. On the translocation of masses. Dokl. Akad.

Nauk SSSR, 37:227-229, 1942.

</aside>

See also

<dt><a class="reference internal" href="#ot.lp.dmmot_monge_1dgrid_optimize" title="ot.lp.dmmot_monge_1dgrid_optimize"><code class="xref py py-obj docutils literal notranslate">ot.lp.dmmot_monge_1dgrid_optimize</code></a></dt><dd>Optimize the d-Dimensional Earth

</dd>

</dl>

<code class="xref py py-obj docutils literal notranslate">Mover</code>

</div>

</dd></dl>

ot.lp.dmmot_monge_1dgrid_optimize(A, niters=100, lr_init=1e-05, lr_decay=0.995, print_rate=100, verbose=False, log=False)<a class="reference internal" href="../_modules/ot/lp/dmmot.html#dmmot_monge_1dgrid_optimize">[source]</a><a class="headerlink" href="#ot.lp.dmmot_monge_1dgrid_optimize" title="Link to this definition"></a></dt>

<dd>Minimize the d-dimensional EMD using gradient descent.

Discrete Multi-Marginal Optimal Transport (d-MMOT): Let \(a_1, \ldots,

a_d\in\mathbb{R}^n_{+}\) be discrete probability distributions. Here,

the d-MMOT is the LP,

\[\begin{split}\begin{align}\begin{aligned}

\underset{x\in\mathbb{R}^{n^{d}}_{+}} {\textrm{min}}

\sum_{i_1,\ldots,i_d} c(i_1,\ldots, i_d)\, x(i_1,\ldots,i_d) \quad

\textrm{s.t.}

\sum_{i_2,\ldots,i_d} x(i_1,\ldots,i_d) &= a_1(i_i),

(\forall i_1\in[n])\\

\qquad\vdots\\

\sum_{i_1,\ldots,i_{d-1}} x(i_1,\ldots,i_d) &= a_{d}(i_{d}),

(\forall i_d\in[n]).

\end{aligned}

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

The dual linear program of the d-MMOT problem is:

\[\underset{z_j\in\mathbb{R}^n, j\in[d]}{\textrm{maximize}}\qquad\sum_{j}

a_j'z_j\qquad \textrm{subject to}\qquad z_{1}(i_1)+\cdots+z_{d}(i_{d})

\leq c(i_1,\ldots,i_{d}),\]</div>

where the indices in the constraints include all $i_j\in[n]$, :math:

<cite>jin[d]</cite>. Denote by $\phi(a_1,\ldots,a_d)$, the optimal objective

value of the LP in d-MMOT problem. Let $z^*$ be an optimal solution

to the dual program. Then,

\[\begin{split}\begin{align}

\nabla \phi(a_1,\ldots,a_{d}) &= z^*,

~~\text{and for any $t\in \mathbb{R}$,}~~

\phi(a_1,a_2,\ldots,a_{d}) = \sum_{j}a_j'

(z_j^* + t\, \eta), \nonumber \\

\text{where } \eta &:= (z_1^{*}(n)\,e, z^*_1(n)\,e, \cdots,

z^*_{d}(n)\,e)

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

Using these dual variables naturally provided by the algorithm in

ot.lp.dmmot_monge_1dgrid_loss, gradient steps move each input distribution

to minimize their d-mmot distance.

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

<li>niters (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional (default=100)) – The maximum number of iterations for the optimization algorithm.</li>

<li>lr_init (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>, optional (default=1e-5)) – The initial learning rate (step size) for the optimization algorithm.</li>

<li>lr_decay (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>, optional (default=0.995)) – The learning rate decay rate in each iteration.</li>

<li>print_rate (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional (default=100)) – The rate at which to print the objective value and gradient norm

during the optimization algorithm.</li>

</ul>

</dd>

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

<li>a (list of ndarrays, each of shape (n,)) – The optimal solution as a list of n approximate barycenters, each of

length vecsize.</li>

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

</ul>

</dd>

</dl>

References

[55]

Ronak Mehta, Jeffery Kline, Vishnu Suresh Lokhande, Glenn Fung, &

Vikas Singh (2023). Efficient Discrete Multi Marginal Optimal

Transport Regularization. In The Eleventh International

Conference on Learning Representations.

</aside>

[60]

Olvi L Mangasarian and RR Meyer. Nonlinear perturbation of linear

programs. SIAM Journal on Control and Optimization, 17(6):745-752, 1979

</aside>

[59]

Michael C Ferris and Olvi L Mangasarian. Finite perturbation of

convex programs. Applied Mathematics and Optimization, 23(1):263-273,

1991.

</aside>

See also

<dt><a class="reference internal" href="#ot.lp.dmmot_monge_1dgrid_loss" title="ot.lp.dmmot_monge_1dgrid_loss"><code class="xref py py-obj docutils literal notranslate">ot.lp.dmmot_monge_1dgrid_loss</code></a></dt><dd>d-Dimensional Earth Mover’s Solver

</dd>

</dl>

</div>

</dd></dl>

ot.lp.emd(a, b, M, numItermax=100000, log=False, center_dual=True, numThreads=1, check_marginals=True, potentials_init=None)<a class="reference internal" href="../_modules/ot/lp/_network_simplex.html#emd">[source]</a><a class="headerlink" href="#ot.lp.emd" title="Link to this definition"></a></dt>

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

\[ \begin{align}\begin{aligned}\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F\\s.t. \ \gamma \mathbf{1} = \mathbf{a}\\ \gamma^T \mathbf{1} = \mathbf{b}\\ \gamma \geq 0\end{aligned}\end{align} \]</div>

where :

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

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

</ul>

Warning

Note that the $\mathbf{M}$ matrix in numpy needs to be a C-order

numpy.array in float64 format. It will be converted if not in this

format

</div>

Note

This function is backend-compatible and will work on arrays

from all compatible backends. But the algorithm uses the C++ CPU backend

which can lead to copy overhead on GPU arrays.

</div>

Note

This function will cast the computed transport plan to the data type

of the provided input with the following priority: $\mathbf{a}$,

then $\mathbf{b}$, then $\mathbf{M}$ if marginals are not provided.

Casting to an integer tensor might result in a loss of precision.

If this behaviour is unwanted, please make sure to provide a

floating point input.

</div>

Note

An error will be raised if the vectors $\mathbf{a}$ and $\mathbf{b}$ do not sum to the same value.

</div>

Uses the algorithm proposed in <a class="reference internal" href="#references-emd">[1]</a>.

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

<li>a ((ns,) array-like, <a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>) – Source histogram (uniform weight if empty list)</li>

<li>b ((nt,) array-like, <a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>) – Target histogram (uniform weight if empty list)</li>

<li>M ((ns,nt) array-like or sparse matrix, <a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>) – Loss matrix. Can be:

<ul>

<li>Dense array (c-order array in numpy with type float64)</li>

<li>Sparse matrix in backend’s format (scipy.sparse.coo_matrix for NumPy backend,

torch.sparse_coo_tensor for PyTorch backend, etc.)</li>

</ul>

</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 (default=100000)) – The maximum number of iterations before stopping the optimization

algorithm if it has not converged.</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 (default=False)) – If True, returns a dictionary containing the cost and dual variables.

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

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

<code class="xref py py-func docutils literal notranslate">ot.lp.center_ot_dual()</code>.</li>

<li>numThreads (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a> or "max", optional (default=1)) – Deprecated compatibility parameter. The network simplex solver no

longer uses OpenMP, so this parameter is ignored.</li>

<li>check_marginals (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#bool" title="(in Python v3.15)">bool</a>, optional (default=True)) – If True, checks that the marginals mass are equal. If False, skips the

check.</li>

<li>potentials_init (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#tuple" title="(in Python v3.15)">tuple</a> of two arrays (alpha, beta), optional (default=None)) – Warmstart dual potentials to accelerate convergence. Should be a tuple

(alpha, beta) where alpha is shape (ns,) and beta is shape (nt,).

These potentials are used to guide initial pivots in the network simplex.

Typically obtained from a previous EMD solve or Sinkhorn approximation.</li>

<li>note: (..) – The solver automatically detects sparse format using the backend’s: <code class="xref py py-meth docutils literal notranslate">issparse()</code> method. For sparse inputs:

<ul>

<li>Uses a memory-efficient sparse EMD algorithm</li>

<li>Returns the transport plan as a sparse matrix in the backend’s format</li>

<li>Supports scipy.sparse (NumPy), torch.sparse (PyTorch), etc.</li>

<li>JAX and TensorFlow backends don’t support sparse matrices</li>

</ul>

</li>

</ul>

</dd>

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

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

<ul>

<li>For dense inputs: returns a dense array</li>

<li>For sparse inputs: returns a sparse matrix in the backend’s format

(e.g., scipy.sparse.coo_matrix for NumPy, torch.sparse_coo_tensor for PyTorch)</li>

</ul>

</li>

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

[1]

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.

</aside>

See also

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

</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.lp.emd2(a, b, M, processes=1, numItermax=100000, log=False, return_matrix=False, center_dual=True, numThreads=1, check_marginals=True, potentials_init=None)<a class="reference internal" href="../_modules/ot/lp/_network_simplex.html#emd2">[source]</a><a class="headerlink" href="#ot.lp.emd2" title="Link to this definition"></a></dt>

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

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

where :

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

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

</ul>

Note

This function is backend-compatible and will work on arrays

from all compatible backends. But the algorithm uses the C++ CPU backend

which can lead to copy overhead on GPU arrays.

</div>

Note

This function will cast the computed transport plan and

transportation loss to the data type of the provided input with the

following priority : $\mathbf{a}$, then $\mathbf{b}$,

then $\mathbf{M}$ if marginals are not provided.

Casting to an integer tensor might result in a loss of precision.

If this behaviour is unwanted, please make sure to provide a

floating point input.

</div>

Note

</div>

Uses the algorithm proposed in <a class="reference internal" href="#references-emd2">[1]</a>.

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

<li>a ((ns,) array-like, float64) – Source histogram (uniform weight if empty list)</li>

<li>b ((nt,) array-like, float64) – Target histogram (uniform weight if empty list)</li>

<li>M ((ns,nt) array-like or sparse matrix, float64) – Loss matrix. Can be:

<ul>

<li>Dense array (c-order array in numpy with type float64)</li>

<li>Sparse matrix in backend’s format (scipy.sparse.coo_matrix for NumPy backend,

torch.sparse_coo_tensor for PyTorch backend, etc.)</li>

</ul>

</li>

<li>processes (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional (default=1)) – Nb of processes used for multiple emd computation (deprecated)</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

<code class="xref py py-func docutils literal notranslate">ot.lp.center_ot_dual()</code>.</li>

longer uses OpenMP, so this parameter is ignored.</li>

check.</li>

(alpha, beta) where alpha is shape (ns,) and beta is shape (nt,).

These potentials are used to guide initial pivots in the network simplex.

Typically obtained from a previous EMD solve or Sinkhorn approximation.</li>

<ul>

<li>Uses a memory-efficient sparse EMD algorithm</li>

<li>Edges not included are treated as having infinite cost (forbidden)</li>

<li>Supports scipy.sparse (NumPy), torch.sparse (PyTorch), etc.</li>

<li>JAX and TensorFlow backends don’t support sparse matrices</li>

</ul>

</li>

</ul>

</dd>

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

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

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

variables (u, v), exit status, and optionally the optimal

transportation matrix (G) if return_matrix is True</li>

</ul>

</dd>

</dl>

Examples

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

perform automatic conversion to numpy arrays

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

0.0

</pre></div>

</div>

References

[1]

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.

</aside>

See also

</dd>

</dl>

</div>

</dd></dl>

ot.lp.emd2_1d(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/solver_1d.html#emd2_1d">[source]</a><a class="headerlink" href="#ot.lp.emd2_1d" title="Link 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>

This implementation only supports metrics

of the form $d(x, y) = |x - y|^p$.

Uses the algorithm detailed in <a href="#id28">[1]_</a>

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

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

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

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

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

<li>metric (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)">str</a>, optional (default='sqeuclidean')) – Metric to be used. Only works with either of the strings

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

<li>p (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>, optional (default=1.0)) – The p-norm to apply for if metric=’minkowski’</li>

<li>dense (boolean, optional (default=True)) – If True, returns $\gamma$ 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

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

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

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

0.5

</pre></div>

</div>

References

[1]

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

Transport”, 2018.

</aside>

See also

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

</dd>

<dt><a class="reference internal" href="#ot.lp.emd_1d" title="ot.lp.emd_1d"><code class="xref py py-obj 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>

ot.lp.emd2_lazy(X_a, X_b, a=None, b=None, metric='sqeuclidean', numItermax=100000, log=False, return_matrix=True, center_dual=True, check_marginals=True, potentials_init=None)<a class="reference internal" href="../_modules/ot/lp/_network_simplex.html#emd2_lazy">[source]</a><a class="headerlink" href="#ot.lp.emd2_lazy" title="Link to this definition"></a></dt>

<dd>Solves the Earth Movers distance problem with lazy cost computation and returns the loss

\[ \begin{align}\begin{aligned}\min_\gamma \quad \langle \gamma, \mathbf{M}(\mathbf{X}_a, \mathbf{X}_b) \rangle_F\\s.t. \ \gamma \mathbf{1} = \mathbf{a}\\ \gamma^T \mathbf{1} = \mathbf{b}\\ \gamma \geq 0\end{aligned}\end{align} \]</div>

where :

<li>$\mathbf{M}(\mathbf{X}_a, \mathbf{X}_b)$ is computed on-the-fly from coordinates</li>

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

</ul>

Note

This function computes distances on-the-fly during the network simplex algorithm,

avoiding the O(ns*nt) memory cost of pre-computing the full cost matrix. Memory usage

is O(ns+nt) instead.

</div>

Note

This function is backend-compatible and will work on arrays

from all compatible backends. But the algorithm uses the C++ CPU backend

which can lead to copy overhead on GPU arrays.

</div>

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

<li>X_a ((ns, dim) array-like, float64) – Source sample coordinates</li>

<li>X_b ((nt, dim) array-like, float64) – Target sample coordinates</li>

<li>a ((ns,) array-like, float64, optional) – Source histogram (uniform weight if None)</li>

<li>b ((nt,) array-like, float64, optional) – Target histogram (uniform weight if None)</li>

<li>metric (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#str" title="(in Python v3.15)">str</a>, optional (default='sqeuclidean')) – Distance metric for cost computation. Options:

<ul>

<li>’sqeuclidean’: Squared Euclidean distance</li>

<li>’euclidean’: Euclidean distance</li>

<li>’cityblock’: Manhattan/L1 distance</li>

</ul>

</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 (default=100000)) – Maximum number of iterations before stopping if not converged</li>

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

and optionally the transport plan (sparse format)</li>

<li>center_dual (boolean, optional (default=True)) – If True, centers the dual potential using <code class="xref py py-func docutils literal notranslate">ot.lp.center_ot_dual()</code></li>

<li>potentials_init (<a class="reference external" href="https://docs.python.org/3/builtins/stdtypes.html#tuple" title="(in Python v3.15)">tuple</a> of (ns,) and (nt,) arrays, optional) – Initial dual potentials (u, v) to warmstart the solver. If provided,

the solver starts from these potentials instead of a cold start.</li>

</ul>

</dd>

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

<li>W (float) – Optimal transportation loss</li>

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

<ul>

<li>cost: the optimal transportation cost</li>

<li>u, v: dual variables</li>

<li>warning: solver status message</li>

<li>result_code: solver return code</li>

<li>G: (optional) sparse transport plan if return_matrix=True</li>

</ul>

</li>

</ul>

</dd>

</dl>

See also

<dt><a class="reference internal" href="../all.html#ot.emd2" title="ot.emd2"><code class="xref py py-obj docutils literal notranslate">ot.emd2</code></a></dt><dd>EMD with pre-computed cost matrix

</dd>

<dt><code class="xref py py-obj docutils literal notranslate">ot.lp.emd_c_lazy</code></dt><dd>Low-level C++ lazy solver

</dd>

</dl>

</div>

</dd></dl>

ot.lp.emd_1d(x_a, x_b, a=None, b=None, metric='sqeuclidean', p=1.0, dense=True, log=False, check_marginals=True)<a class="reference internal" href="../_modules/ot/lp/solver_1d.html#emd_1d">[source]</a><a class="headerlink" href="#ot.lp.emd_1d" title="Link 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>

This implementation only supports metrics

of the form $d(x, y) = |x - y|^p$.

Uses the algorithm detailed in <a href="#id29">[1]_</a>

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

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

<li>p (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#float" title="(in Python v3.15)">float</a>, optional (default=1.0)) – The p-norm to apply for if metric=’minkowski’</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>

check.</li>

</ul>

</dd>

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

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

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

of the non-zero elements of the transportation matrix</li>

</ul>

</dd>

</dl>

Examples

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

performs automatic conversion to numpy arrays

>>> 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

[1]

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

Transport”, 2018.

</aside>

See also

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

</dd>

<dt><a class="reference internal" href="#ot.lp.emd2_1d" title="ot.lp.emd2_1d"><code class="xref py py-obj 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>

ot.lp.emd_1d_dual_backprop(u_values, v_values, u_weights=None, v_weights=None, p=1, require_sort=True)<a class="reference internal" href="../_modules/ot/lp/solver_1d.html#emd_1d_dual_backprop">[source]</a><a class="headerlink" href="#ot.lp.emd_1d_dual_backprop" title="Link to this definition"></a></dt>

<dd>Computes the 1 dimensional OT loss between two (batched) empirical

distributions

\[OT_{loss} = \int_0^1 |cdf_u^{-1}(q) - cdf_v^{-1}(q)|^p dq\]</div>

and returns the dual potentials and the loss, i.e. such that

\[OT_{loss}(u,v) = \int f(x)\mathrm{d}u(x) + \int g(y)\mathrm{d}v(y).\]</div>

Warning

This function only works in pytorch or jax as it backpropagates through the <cite>wasserstein_1d</cite> function.

</div>

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

<li>u_values (array-like, shape (n, ...)) – locations of the first empirical distribution</li>

<li>v_values (array-like, shape (m, ...)) – locations of the second empirical distribution</li>

<li>u_weights (array-like, shape (n, ...), optional) – weights of the first empirical distribution, if None then uniform weights are used</li>

<li>v_weights (array-like, shape (m, ...), optional) – weights of the second empirical distribution, if None then uniform weights are used</li>

<li>p (<a class="reference external" href="https://docs.python.org/3/builtins/functions.html#int" title="(in Python v3.15)">int</a>, optional) – order of the ground metric used, should be at least 1, default is 1</li>

the function, default is True</li>

</ul>

</dd>

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

<li>f (array-like shape (n, …)) – First dual potential</li>

<li>g (array-like shape (m, …)) – Second dual potential</li>

<li>loss (float/array-like, shape (…)) – the batched EMD</li>

</ul>

</dd>

</dl>

</dd></dl>

ot.lp.emd_1d_sorted(u_weights, v_weights, u, v, metric='sqeuclidean', p=1.0)<a class="headerlink" href="#ot.lp.emd_1d_sorted" title="Link to this definition"></a></dt>

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