ot.lp

Solvers for the original linear program OT problem.

ot.lp.NorthWestMMGluing(pi_list, a=None, log=False, nx=None)[source]

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 J of shape (N, K) where each J[i] is of the form (J[i, 1], …, J[i, K]) with each J[i, k] between 0 and \(n_k-1\), and a weight vector w of size N, such that the K-plan \(\rho\) writes:

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

This representation is useful for its memory efficiency, as it avoids storing the full K-marginal plan.

If log=True, 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.

Parameters:

• pi_list (list of arrays (n, n_k)) – List of transport plans.

• log (bool, optional) – If True, return a log dictionary (computationally expensive).

• 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.

• nx (backend, optional) – The backend to use. If None is provided, the backend of the first transport plan is used.

Returns:

• J (array (N, K)) – The indices (J[i, 1], …, J[i, K]) of the K-plan \(\rho\).

• w (array (N,)) – The weights w_i of the K-plan \(\rho\).

• log_dict (dict, optional) – If log=True, a dictionary containing the full K+1-marginal transport plan under the key ‘gamma’.

ot.lp.barycenter(A, M, weights=None, verbose=False, log=False, solver='highs-ipm')[source]

Compute the Wasserstein barycenter of distributions A

The function solves the following optimization problem [16]:

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

where :

• \(W_1(\cdot,\cdot)\) is the Wasserstein distance (see ot.emd.sinkhorn)

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

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).

Parameters:

• A (np.ndarray (d,n)) – n training distributions a_i of size d

• M (np.ndarray (d,d)) – loss matrix for OT

• reg (float) – Regularization term >0

• weights (np.ndarray (n,)) – Weights of each histogram a_i on the simplex (barycentric coordinates)

• verbose (bool, optional) – Print information along iterations

• log (bool, optional) – record log if True

• 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.

Returns:

• a ((d,) ndarray) – Wasserstein barycenter

• log (dict) – log dictionary return only if log==True in parameters

References

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)[source]

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.

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

where:

• \(F_u\) and \(F_v\) are respectively the cdfs of \(u\) and \(v\)

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

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

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

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

Parameters:

• u_values (ndarray, shape (n, ...)) – samples in the source domain (coordinates on [0,1[)

• v_values (ndarray, shape (n, ...)) – samples in the target domain (coordinates on [0,1[)

• u_weights (ndarray, shape (n, ...), optional) – samples weights in the source domain

• v_weights (ndarray, shape (n, ...), optional) – samples weights in the target domain

• p (float, optional (default=1)) – Power p used for computing the Wasserstein distance

• Lm (int, optional) – Lower bound dC

• Lp (int, optional) – Upper bound dC

• tm (float, optional) – Lower bound theta

• tp (float, optional) – Upper bound theta

• eps (float, optional) – Stopping condition

• require_sort (bool, optional) – If True, sort the values.

• log (bool, optional) – If True, returns also the optimal theta

Returns:

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

• log (dict, optional) – log dictionary returned only if log==True in parameters

Examples

>>> 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])

References

ot.lp.check_number_threads(numThreads)[source]

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

Parameters:

numThreads (int or str) – The requested number of threads, should either be a strictly positive integer or “max” or None

Returns:

numThreads – Corrected number of threads

Return type:

int

ot.lp.dmmot_monge_1dgrid_loss(A, verbose=False, log=False)[source]

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]:

\[\begin{split}\begin{align}\begin{aligned} \underset{\gamma\in\mathbb{R}^{n^{d}}_{+}} {\textrm{min}} \sum_{i_1,\ldots,i_d} c(i_1,\ldots, i_d)\, \gamma(i_1,\ldots,i_d) \quad \textrm{s.t.} \sum_{i_2,\ldots,i_d} \gamma(i_1,\ldots,i_d) &= a_1(i_i), (\forall i_1\in[n])\\ \qquad\vdots\\ \sum_{i_1,\ldots,i_{d-1}} \gamma(i_1,\ldots,i_d) &= a_{d}(i_{d}), (\forall i_d\in[n]). \end{aligned} \end{align}\end{split}\]

Parameters:

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

• verbose (bool, optional) – If True, print debugging information during execution. Default=False.

• log (bool, optional) – If True, record log. Default is False.

Returns:

• obj (float) – the value of the primal objective function evaluated at the solution.

• 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.

References

See also

ot.lp.dmmot_monge_1dgrid_optimize

Optimize the d-Dimensional Earth

Mover

ot.lp.dmmot_monge_1dgrid_optimize(A, niters=100, lr_init=1e-05, lr_decay=0.995, print_rate=100, verbose=False, log=False)[source]

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,

\[\begin{split}\begin{align}\begin{aligned} \underset{x\in\mathbb{R}^{n^{d}}_{+}} {\textrm{min}} \sum_{i_1,\ldots,i_d} c(i_1,\ldots, i_d)\, x(i_1,\ldots,i_d) \quad \textrm{s.t.} \sum_{i_2,\ldots,i_d} x(i_1,\ldots,i_d) &= a_1(i_i), (\forall i_1\in[n])\\ \qquad\vdots\\ \sum_{i_1,\ldots,i_{d-1}} x(i_1,\ldots,i_d) &= a_{d}(i_{d}), (\forall i_d\in[n]). \end{aligned} \end{align}\end{split}\]

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

\[\underset{z_j\in\mathbb{R}^n, j\in[d]}{\textrm{maximize}}\qquad\sum_{j} a_j'z_j\qquad \textrm{subject to}\qquad z_{1}(i_1)+\cdots+z_{d}(i_{d}) \leq c(i_1,\ldots,i_{d}),\]

where the indices in the constraints include all \(i_j\in[n]\), :math: jin[d]. 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,

\[\begin{split}\begin{align} \nabla \phi(a_1,\ldots,a_{d}) &= z^*, ~~\text{and for any $t\in \mathbb{R}$,}~~ \phi(a_1,a_2,\ldots,a_{d}) = \sum_{j}a_j' (z_j^* + t\, \eta), \nonumber \\ \text{where } \eta &:= (z_1^{*}(n)\,e, z^*_1(n)\,e, \cdots, z^*_{d}(n)\,e) \end{align}\end{split}\]

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.

Parameters:

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

• niters (int, optional (default=100)) – The maximum number of iterations for the optimization algorithm.

• lr_init (float, optional (default=1e-5)) – The initial learning rate (step size) for the optimization algorithm.

• lr_decay (float, optional (default=0.995)) – The learning rate decay rate in each iteration.

• print_rate (int, optional (default=100)) – The rate at which to print the objective value and gradient norm during the optimization algorithm.

• verbose (bool, optional) – If True, print debugging information during execution. Default=False.

• log (bool, optional) – If True, record log. Default is False.

Returns:

• a (list of ndarrays, each of shape (n,)) – The optimal solution as a list of n approximate barycenters, each of length vecsize.

• log (dict) – log dictionary return only if log==True in parameters

References

See also

ot.lp.dmmot_monge_1dgrid_loss

d-Dimensional Earth Mover’s Solver

ot.lp.emd(a, b, M, numItermax=100000, log=False, center_dual=True, numThreads=1, check_marginals=True)[source]

Solves the Earth Movers distance problem and returns the OT matrix

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

where :

• \(\mathbf{M}\) is the metric cost matrix

• \(\mathbf{a}\) and \(\mathbf{b}\) are the sample weights

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

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.

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.

Note

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

Uses the algorithm proposed in [1].

Parameters:

• a ((ns,) array-like, float) – Source histogram (uniform weight if empty list)

• b ((nt,) array-like, float) – Target histogram (uniform weight if empty list)

• M ((ns,nt) array-like, float) – Loss matrix (c-order array in numpy with type float64)

• numItermax (int, optional (default=100000)) – The maximum number of iterations before stopping the optimization algorithm if it has not converged.

• log (bool, optional (default=False)) – If True, returns a dictionary containing the cost and dual variables. Otherwise returns only the optimal transportation matrix.

• center_dual (boolean, optional (default=True)) – If True, centers the dual potential using function ot.lp.center_ot_dual().

• numThreads (int or "max", optional (default=1, i.e. OpenMP is not used)) – If compiled with OpenMP, chooses the number of threads to parallelize. “max” selects the highest number possible.

• check_marginals (bool, optional (default=True)) – If True, checks that the marginals mass are equal. If False, skips the check.

Returns:

• gamma (array-like, shape (ns, nt)) – Optimal transportation matrix for the given parameters

• log (dict, optional) – If input log is true, a dictionary containing the cost and dual variables and exit status

Examples

Simple example with obvious solution. The function emd accepts lists and perform automatic conversion to numpy arrays

>>> 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]])

References

See also

ot.bregman.sinkhorn

Entropic regularized OT

ot.optim.cg

General regularized OT

ot.lp.emd2(a, b, M, processes=1, numItermax=100000, log=False, return_matrix=False, center_dual=True, numThreads=1, check_marginals=True)[source]

Solves the Earth Movers distance problem and returns the loss

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

where :

• \(\mathbf{M}\) is the metric cost matrix

• \(\mathbf{a}\) and \(\mathbf{b}\) are the sample weights

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.

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.

Note

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

Uses the algorithm proposed in [1].

Parameters:

• a ((ns,) array-like, float64) – Source histogram (uniform weight if empty list)

• b ((nt,) array-like, float64) – Target histogram (uniform weight if empty list)

• M ((ns,nt) array-like, float64) – Loss matrix (for numpy c-order array with type float64)

• processes (int, optional (default=1)) – Nb of processes used for multiple emd computation (deprecated)

• numItermax (int, optional (default=100000)) – The maximum number of iterations before stopping the optimization algorithm if it has not converged.

• log (boolean, optional (default=False)) – If True, returns a dictionary containing dual variables. Otherwise returns only the optimal transportation cost.

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

• center_dual (boolean, optional (default=True)) – If True, centers the dual potential using function ot.lp.center_ot_dual().

• numThreads (int or "max", optional (default=1, i.e. OpenMP is not used)) – If compiled with OpenMP, chooses the number of threads to parallelize. “max” selects the highest number possible.

• check_marginals (bool, optional (default=True)) – If True, checks that the marginals mass are equal. If False, skips the check.

Returns:

• W (float, array-like) – Optimal transportation loss for the given parameters

• log (dict) – If input log is true, a dictionary containing dual variables and exit status

Examples

Simple example with obvious solution. The function emd accepts lists and perform automatic conversion to numpy arrays

>>> import ot
>>> a=[.5,.5]
>>> b=[.5,.5]
>>> M=[[0.,1.],[1.,0.]]
>>> ot.emd2(a,b,M)
0.0

References

See also

ot.bregman.sinkhorn

Entropic regularized OT

ot.optim.cg

General regularized OT

ot.lp.emd2_1d(x_a, x_b, a=None, b=None, metric='sqeuclidean', p=1.0, dense=True, log=False)[source]

Solves the Earth Movers distance problem between 1d measures and returns the loss

\[ \begin{align}\begin{aligned}\gamma = arg\min_\gamma \sum_i \sum_j \gamma_{ij} d(x_a[i], x_b[j])\\s.t. \gamma 1 = a, \gamma^T 1= b, \gamma\geq 0\end{aligned}\end{align} \]

where :

• d is the metric

• \(x_a\) and \(x_b\) are the samples

• a and b are the sample weights

This implementation only supports metrics of the form \(d(x, y) = |x - y|^p\).

Uses the algorithm detailed in [1]_

Parameters:

• x_a (ndarray of float64, shape (ns,) or (ns, 1)) – Source dirac locations (on the real line)

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

• a (ndarray of float64, shape (ns,), optional) – Source histogram (default is uniform weight)

• b (ndarray of float64, shape (nt,), optional) – Target histogram (default is uniform weight)

• metric (str, optional (default='sqeuclidean')) – Metric to be used. Only works with either of the strings ‘sqeuclidean’, ‘minkowski’, ‘cityblock’, or ‘euclidean’.

• p (float, optional (default=1.0)) – The p-norm to apply for if metric=’minkowski’

• 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 coo_matrix format. Only used if log is set to True. Due to implementation details, this function runs faster when dense is set to False.

• log (boolean, optional (default=False)) – If True, returns a dictionary containing the transportation matrix. Otherwise returns only the loss.

Returns:

• loss (float) – Cost associated to the optimal transportation

• log (dict) – If input log is True, a dictionary containing the Optimal transportation matrix for the given parameters

Examples

Simple example with obvious solution. The function emd2_1d accepts lists and performs automatic conversion to numpy arrays

>>> import ot
>>> a=[.5, .5]
>>> b=[.5, .5]
>>> x_a = [2., 0.]
>>> x_b = [0., 3.]
>>> ot.emd2_1d(x_a, x_b, a, b)
0.5
>>> ot.emd2_1d(x_a, x_b)
0.5

References

See also

ot.lp.emd2

EMD for multidimensional distributions

ot.lp.emd_1d

EMD for 1d distributions (returns the transportation matrix instead of the cost)

ot.lp.emd_1d(x_a, x_b, a=None, b=None, metric='sqeuclidean', p=1.0, dense=True, log=False, check_marginals=True)[source]

Solves the Earth Movers distance problem between 1d measures and returns the OT matrix

\[ \begin{align}\begin{aligned}\gamma = arg\min_\gamma \sum_i \sum_j \gamma_{ij} d(x_a[i], x_b[j])\\s.t. \gamma 1 = a, \gamma^T 1= b, \gamma\geq 0\end{aligned}\end{align} \]

where :

• d is the metric

• \(x_a\) and \(x_b\) are the samples

• a and b are the sample weights

This implementation only supports metrics of the form \(d(x, y) = |x - y|^p\).

Uses the algorithm detailed in [1]_

Parameters:

• x_a (ndarray of float64, shape (ns,) or (ns, 1)) – Source dirac locations (on the real line)

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

• a (ndarray of float64, shape (ns,), optional) – Source histogram (default is uniform weight)

• b (ndarray of float64, shape (nt,), optional) – Target histogram (default is uniform weight)

• metric (str, optional (default='sqeuclidean')) – Metric to be used. Only works with either of the strings ‘sqeuclidean’, ‘minkowski’, ‘cityblock’, or ‘euclidean’.

• p (float, optional (default=1.0)) – The p-norm to apply for if metric=’minkowski’

• 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 coo_matrix format. Due to implementation details, this function runs faster when ‘sqeuclidean’, ‘minkowski’, ‘cityblock’, or ‘euclidean’ metrics are used.

• log (boolean, optional (default=False)) – If True, returns a dictionary containing the cost. Otherwise returns only the optimal transportation matrix.

• check_marginals (bool, optional (default=True)) – If True, checks that the marginals mass are equal. If False, skips the check.

Returns:

• gamma (ndarray, shape (ns, nt)) – Optimal transportation matrix for the given parameters

• log (dict) – If input log is True, a dictionary containing the cost

Examples

Simple example with obvious solution. The function emd_1d accepts lists and performs automatic conversion to numpy arrays

>>> import ot
>>> a=[.5, .5]
>>> b=[.5, .5]
>>> x_a = [2., 0.]
>>> x_b = [0., 3.]
>>> ot.emd_1d(x_a, x_b, a, b)
array([[0. , 0.5],
       [0.5, 0. ]])
>>> ot.emd_1d(x_a, x_b)
array([[0. , 0.5],
       [0.5, 0. ]])

References

See also

ot.lp.emd

EMD for multidimensional distributions

ot.lp.emd2_1d

EMD for 1d distributions (returns cost instead of the transportation matrix)

ot.lp.emd_1d_sorted(u_weights, v_weights, u, v, metric='sqeuclidean', p=1.0)

Solves the Earth Movers distance problem between sorted 1d measures and returns the OT matrix and the associated cost

Parameters:

• u_weights ((ns,) ndarray, float64) – Source histogram

• v_weights ((nt,) ndarray, float64) – Target histogram

• u ((ns,) ndarray, float64) – Source dirac locations (on the real line)

• v ((nt,) ndarray, float64) – Target dirac locations (on the real line)

• metric (str, optional (default='sqeuclidean')) – Metric to be used. Only works with either of the strings ‘sqeuclidean’, ‘minkowski’, ‘cityblock’, or ‘euclidean’.

• p (float, optional (default=1.0)) – The p-norm to apply for if metric=’minkowski’

Returns:

• gamma ((n, ) ndarray, float64) – Values in the Optimal transportation matrix

• indices ((n, 2) ndarray, int64) – Indices of the values stored in gamma for the Optimal transportation matrix

• cost – cost associated to the optimal transportation

ot.lp.free_support_barycenter(measures_locations, measures_weights, X_init, b=None, weights=None, numItermax=100, stopThr=1e-07, verbose=False, log=None, numThreads=1)[source]

Solves the free support (locations of the barycenters are optimized, not the weights) Wasserstein barycenter problem (i.e. the weighted Frechet mean for the 2-Wasserstein distance), formally:

\[\min_\mathbf{X} \quad \sum_{i=1}^N w_i W_2^2(\mathbf{b}, \mathbf{X}, \mathbf{a}_i, \mathbf{X}_i)\]

where :

• \(w \in \mathbb{(0, 1)}^{N}\)’s are the barycenter weights and sum to one

• measure_weights denotes the \(\mathbf{a}_i \in \mathbb{R}^{k_i}\): empirical measures weights (on simplex)

• measures_locations denotes the \(\mathbf{X}_i \in \mathbb{R}^{k_i, d}\): empirical measures atoms locations

• \(\mathbf{b} \in \mathbb{R}^{k}\) is the desired weights vector of the barycenter

This problem is considered in [20] (Algorithm 2). There are two differences with the following codes:

• we do not optimize over the weights

• we do not do line search for the locations updates, we use i.e. \(\theta = 1\) in [20] (Algorithm 2). This can be seen as a discrete implementation of the fixed-point algorithm of [43] proposed in the continuous setting.

Parameters:

• measures_locations (list of N (k_i,d) array-like) – The discrete support of a measure supported on \(k_i\) locations of a d-dimensional space (\(k_i\) can be different for each element of the list)

• measures_weights (list of N (k_i,) array-like) – Numpy arrays where each numpy array has \(k_i\) non-negatives values summing to one representing the weights of each discrete input measure

• X_init ((k,d) array-like) – Initialization of the support locations (on k atoms) of the barycenter

• b ((k,) array-like) – Initialization of the weights of the barycenter (non-negatives, sum to 1)

• weights ((N,) array-like) – Initialization of the coefficients of the barycenter (non-negatives, sum to 1)

• numItermax (int, optional) – Max number of iterations

• stopThr (float, optional) – Stop threshold on error (>0)

• verbose (bool, optional) – Print information along iterations

• log (bool, optional) – record log if True

• numThreads (int or "max", optional (default=1, i.e. OpenMP is not used)) – If compiled with OpenMP, chooses the number of threads to parallelize. “max” selects the highest number possible.

Returns:

• X ((k,d) array-like) – Support locations (on k atoms) of the barycenter

• .. _references-free-support-barycenter

References

ot.lp.free_support_barycenter_generic_costs(measure_locations, measure_weights, X_init, cost_list, ground_bary=None, a=None, numItermax=100, method='L2_barycentric_proj', stopThr=1e-05, log=False, ground_bary_lr=0.01, ground_bary_numItermax=100, ground_bary_stopThr=1e-05, ground_bary_solver='SGD', clean_measure=False)[source]

Solves the OT barycenter problem [77] for generic costs using the fixed point algorithm, iterating the ground barycenter function B on transport plans between the current barycenter and the measures.

The problem finds an optimal barycenter support X of given size (n, d) (enforced by the initialisation), minimising a sum of pairwise transport costs for the costs \(c_k\):

\[\min_{X} \sum_{k=1}^K \mathcal{T}_{c_k}(X, a, Y_k, b_k),\]

where:

• \(X\) (n, d) is the barycenter support,

• \(a\) (n) is the (fixed) barycenter weights,

• \(Y_k\) (m_k, d_k) is the k-th measure support (measure_locations[k]),

• \(b_k\) (m_k) is the k-th measure weights (measure_weights[k]),

• \(c_k: \mathbb{R}^{n\times d}\times\mathbb{R}^{m_k\times d_k}\rightarrow \mathbb{R}_+^{n\times m_k}\) is the k-th cost function (which computes the pairwise cost matrix)

• \(\mathcal{T}_{c_k}(X, a, Y_k, b)\) is the OT cost between the barycenter measure and the k-th measure with respect to the cost \(c_k\):

\[ \begin{align}\begin{aligned}\mathcal{T}_{c_k}(X, a, Y_k, b_k) = \min_\pi \quad \langle \pi, c_k(X, Y_k) \rangle_F\\s.t. \ \pi \mathbf{1} = \mathbf{a}\\ \pi^T \mathbf{1} = \mathbf{b_k}\\ \pi \geq 0\end{aligned}\end{align} \]

in other words, \(\mathcal{T}_{c_k}(X, a, Y_k, b_k)\) is ot.emd2(a, b_k, c_k(X, Y_k)).

The function \(B:\mathbb{R}^{n\times d_1}\times \cdots\times\mathbb{R}^{n\times d_K} \longrightarrow \mathbb{R}^{n\times d}\) is an input to the solver. B computes solutions of the following minimisation problem given \((Y_1, \cdots, Y_K) \in \mathbb{R}^{n\times d_1}\times\cdots\times\mathbb{R}^{n\times d_K}\) (broadcasted along n):

\[B(y_1, \cdots, y_K) = \mathrm{argmin}_{x \in \mathbb{R}^d} \sum_{k=1}^K c_k(x, y_k),\]

The input function B takes a list of K arrays of shape (n, d_k) and returns an array of shape (n, d). For certain costs, \(B\) can be computed explicitly, or through a numerical solver.

This function implements two algorithms:

• Algorithm 2 from [77] when method=true_fixed_point is used, which may increase the support size of the barycenter at each iteration, with a maximum final size of \(N_0 + T\sum_k n_k - TK\) for T iterations and an initial support size of \(N_0\). The computation of the iterates is done using the North West Corner multi-marginal gluing method. This method has convergence guarantees [77].

• Algorithm 3 from [77] when method=L2_barycentric_proj is used, which is a heuristic simplification which fixes the weights and support size of the barycenter by performing barycentric projections of the pair-wise OT matrices. This method is substantially faster than the first one, but does not have convergence guarantees. (Default)

The implemented methods ([77] Algorithms 2 and 3), generalises [20] and [43] to general costs and includes convergence guarantees, including for discrete measures.

Parameters:

• measure_locations (list of array-like) – List of K arrays of measure positions, each of shape (m_k, d_k).

• measure_weights (list of array-like) – List of K arrays of measure weights, each of shape (m_k).

• X_init (array-like) – Array of shape (n, d) representing initial barycenter points.

• cost_list (list of callable or callable) – List of K cost functions \(c_k: \mathbb{R}^{n\times d}\times\mathbb{R}^{m_k\times d_k} \rightarrow \mathbb{R}_+^{n\times m_k}\). If cost_list is a single callable, the same cost is used K times.

• ground_bary (callable or None, optional) – Function List(array(n, d_k)) -> array(n, d) accepting a list of K arrays of shape \((n\times d_K)\), computing the ground barycenters (broadcasted over n). If not provided, done with Adam on PyTorch (requires PyTorch backend), inefficiently using the cost functions in cost_list.

• a (array-like, optional) – Array of shape (n,) representing weights of the barycenter measure.Defaults to uniform.

• numItermax (int, optional) – Maximum number of iterations (default is 100).

• method (str, optional) – Barycentre method: ‘L2_barycentric_proj’ (default) for Euclidean barycentric projection, or ‘true_fixed_point’ for iterates using the North West Corner multi-marginal gluing method.

• stopThr (float, optional) – If \(W_2^2(a_t, X_t, a_{t+1}, X_{t+1}) < \mathrm{stopThr} \times \frac{1}{n}\|X_t\|_2^2\), terminate (default is 1e-5).

• log (bool, optional) – Whether to return the log dictionary (default is False).

• ground_bary_lr (float, optional) – Learning rate for the ground barycenter solver (if auto is used).

• ground_bary_numItermax (int, optional) – Maximum number of iterations for the ground barycenter solver (if auto is used).

• ground_bary_stopThr (float, optional) – Stop threshold for the ground barycenter solver (if auto is used): stop if the energy decreases less than this value.

• ground_bary_solver (str, optional) – Solver for auto ground bary solver (torch SGD or Adam). Default is “SGD”.

• clean_measure (bool, optional) – For method==’true_fixed_point’, whether to clean the discrete measure (X, a) at each iteration to remove duplicate points and sum their weights (default is False).

Returns:

• X (array-like) – Array of shape (n, d) representing barycenter points.

• log_dict (list of array-like, optional) – log containing the exit status, list of iterations and list of displacements if log is True.

References

Note

For the case of the L2 cost \(c_k(x, y) = \|x-y\|_2^2\), the ground barycenter is simply the Euclidean barycenter, i.e. \(B(y_1, \cdots, y_K) = \sum_k w_k y_k\). In this case, we recover the free-support algorithm from [20].

See also

ot.lp.free_support_barycenter : Free support solver for the case where \(c_k(x,y) = \lambda_k\|x-y\|_2^2\).

ot.lp.generalized_free_support_barycenter : Free support solver for the case where \(c_k(x,y) = \|P_kx-y\|_2^2\) with \(P_k\) linear.

ot.lp.NorthWestMMGluing : gluing method used in the true_fixed_point method.

ot.lp.generalized_free_support_barycenter(X_list, a_list, P_list, n_samples_bary, Y_init=None, b=None, weights=None, numItermax=100, stopThr=1e-07, verbose=False, log=None, numThreads=1, eps=0)[source]

Solves the free support generalized Wasserstein barycenter problem: finding a barycenter (a discrete measure with a fixed amount of points of uniform weights) whose respective projections fit the input measures. More formally:

\[\min_\gamma \quad \sum_{i=1}^p w_i W_2^2(\nu_i, \mathbf{P}_i\#\gamma)\]

where :

• \(\gamma = \sum_{l=1}^n b_l\delta_{y_l}\) is the desired barycenter with each \(y_l \in \mathbb{R}^d\)

• \(\mathbf{b} \in \mathbb{R}^{n}\) is the desired weights vector of the barycenter

• The input measures are \(\nu_i = \sum_{j=1}^{k_i}a_{i,j}\delta_{x_{i,j}}\)

• The \(\mathbf{a}_i \in \mathbb{R}^{k_i}\) are the respective empirical measures weights (on the simplex)

• The \(\mathbf{X}_i \in \mathbb{R}^{k_i, d_i}\) are the respective empirical measures atoms locations

• \(w = (w_1, \cdots w_p)\) are the barycenter coefficients (on the simplex)

• Each \(\mathbf{P}_i \in \mathbb{R}^{d, d_i}\), and \(P_i\#\nu_i = \sum_{j=1}^{k_i}a_{i,j}\delta_{P_ix_{i,j}}\)

As show by [42], this problem can be re-written as a Wasserstein Barycenter problem, which we solve using the free support method [20] (Algorithm 2).

Parameters:

• X_list (list of p (k_i,d_i) array-like) – Discrete supports of the input measures: each consists of \(k_i\) locations of a d_i-dimensional space (\(k_i\) can be different for each element of the list)

• a_list (list of p (k_i,) array-like) – Measure weights: each element is a vector (k_i) on the simplex

• P_list (list of p (d_i,d) array-like) – Each \(P_i\) is a linear map \(\mathbb{R}^{d} \rightarrow \mathbb{R}^{d_i}\)

• n_samples_bary (int) – Number of barycenter points

• Y_init ((n_samples_bary,d) array-like) – Initialization of the support locations (on k atoms) of the barycenter

• b ((n_samples_bary,) array-like) – Initialization of the weights of the barycenter measure (on the simplex)

• weights ((p,) array-like) – Initialization of the coefficients of the barycenter (on the simplex)

• numItermax (int, optional) – Max number of iterations

• stopThr (float, optional) – Stop threshold on error (>0)

• verbose (bool, optional) – Print information along iterations

• log (bool, optional) – record log if True

• numThreads (int or "max", optional (default=1, i.e. OpenMP is not used)) – If compiled with OpenMP, chooses the number of threads to parallelize. “max” selects the highest number possible.

• eps (Stability coefficient for the change of variable matrix inversion) – If the \(\mathbf{P}_i^T\) matrices don’t span \(\mathbb{R}^d\), the problem is ill-defined and a matrix inversion will fail. In this case one may set eps=1e-8 and get a solution anyway (which may make little sense)

Returns:

Y – Support locations (on n_samples_bary atoms) of the barycenter

Return type:

(n_samples_bary,d) array-like

References

ot.lp.linear_circular_ot(u_values, v_values=None, u_weights=None, v_weights=None)[source]

Computes the Linear Circular Optimal Transport distance from [78] using \(\eta=\mathrm{Unif}(S^1)\) as reference measure. 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.

General loss returned:

\[\mathrm{LCOT}_2^2(\mu, \nu) = \int_0^1 d_{S^1}\big(\hat{\mu}(t), \hat{\nu}(t)\big)^2\ \mathrm{d}t\]

where \(\hat{\mu}(x)=F_{\mu}^{-1}(x-\int z\mathrm{d}\mu(z)+\frac12) - x\) for all \(x\in [0,1[\), and \(d_{S^1}(x,y)=\min(|x-y|, 1-|x-y|)\) for \(x,y\in [0,1[\).

Parameters:

• u_values (ndarray, shape (n, ...)) – samples in the source domain (coordinates on [0,1[)

• v_values (ndarray, shape (n, ...), optional) – samples in the target domain (coordinates on [0,1[), if None, compute distance against uniform distribution

• u_weights (ndarray, shape (n, ...), optional) – samples weights in the source domain

• v_weights (ndarray, shape (n, ...), optional) – samples weights in the target domain

Returns:

loss – Batched cost associated to the linear optimal transportation

Return type:

float/array-like, shape (…)

Examples

>>> u = np.array([[0.2,0.5,0.8]])%1
>>> v = np.array([[0.4,0.5,0.7]])%1
>>> linear_circular_ot(u.T, v.T)
array([0.0127])

References

ot.lp.ot_barycenter_energy(measure_locations, measure_weights, X, a, cost_list, nx=None)[source]

Computes the energy of the OT barycenter functional for a given barycenter support X and weights a: .. math:

V(X, a) = \sum_{k=1}^K w_k \mathcal{T}_{c_k}(X, a, Y_k, b_k),

where: - \(X\) (n, d) is the barycenter support, - \(a\) (n) is the barycenter weights, - \(Y_k\) (m_k, d_k) is the k-th measure support

(measure_locations[k]),

• \(b_k\) (m_k) is the k-th measure weights (measure_weights[k]),

• :math:`c_k: mathbb{R}^{ntimes d}timesmathbb{R}^{m_ktimes d_k}

  rightarrow mathbb{R}_+^{ntimes m_k}` is the k-th cost function (which computes the pairwise cost matrix)

• \(\mathcal{T}_{c_k}(X, a, Y_k, b)\) is the OT cost between the barycenter measure and the k-th measure with respect to the cost \(c_k\).

The function computes \(V(X, a)\) as defined above.

Parameters:

• measure_locations (list of array-like) – List of K arrays of measure positions, each of shape (m_k, d_k).

• measure_weights (list of array-like) – List of K arrays of measure weights, each of shape (m_k).

• X (array-like) – Array of shape (n, d) representing barycenter points.

• a (array-like) – Array of shape (n,) representing barycenter weights.

• cost_list (list of callable or callable) – List of K cost functions \(c_k: \mathbb{R}^{n\times d}\times\mathbb{R}^{m_k\times d_k} \rightarrow \mathbb{R}_+^{n\times m_k}\). If cost_list is a single callable, the same cost is used K times.

• nx (backend, optional) – The backend to use.

Returns:

V – The value of the OT barycenter functional \(V(X, a)\).

Return type:

float

References

See also

ot.lp.free_support_barycenter_generic_costs

Free support solver for the

associated

ot.lp.semidiscrete_wasserstein2_unif_circle(u_values, u_weights=None)[source]

Computes the closed-form for the 2-Wasserstein distance between samples and a uniform distribution on \(S^1\) 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.

\[W_2^2(\mu_n, \nu) = \sum_{i=1}^n \alpha_i x_i^2 - \left(\sum_{i=1}^n \alpha_i x_i\right)^2 + \sum_{i=1}^n \alpha_i x_i \left(1-\alpha_i-2\sum_{k=1}^{i-1}\alpha_k\right) + \frac{1}{12}\]

where:

• \(\nu=\mathrm{Unif}(S^1)\) and \(\mu_n = \sum_{i=1}^n \alpha_i \delta_{x_i}\)

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

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

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

Parameters:

• u_values (ndarray, shape (n, ...)) – Samples

• u_weights (ndarray, shape (n, ...), optional) – samples weights in the source domain

Returns:

loss – Batched cost associated to the optimal transportation

Return type:

float/array-like, shape (…)

Examples

>>> x0 = np.array([[0], [0.2], [0.4]])
>>> semidiscrete_wasserstein2_unif_circle(x0)
array([0.02111111])

References

ot.lp.wasserstein_1d(u_values, v_values, u_weights=None, v_weights=None, p=1, require_sort=True)[source]

Computes the 1 dimensional OT loss [15] between two (batched) empirical distributions

It is formally the p-Wasserstein distance raised to the power p. We do so in a vectorized way by first building the individual quantile functions then integrating them.

This function should be preferred to emd_1d whenever the backend is different to numpy, and when gradients over either sample positions or weights are required.

Parameters:

• u_values (array-like, shape (n, ...)) – locations of the first empirical distribution

• v_values (array-like, shape (m, ...)) – locations of the second empirical distribution

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

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

• p (int, optional) – order of the ground metric used, should be at least 1 (see [2, Chap. 2], default is 1

• require_sort (bool, optional) – sort the distributions atoms locations, if False we will consider they have been sorted prior to being passed to the function, default is True

Returns:

cost – the batched EMD

Return type:

float/array-like, shape (…)

References

ot.lp.wasserstein_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)[source]

Computes the Wasserstein distance on the circle using either [45] for p=1 or the binary search algorithm proposed in [44] otherwise. 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 requires to first find the coordinates using e.g. the atan2 function.

General loss returned:

\[OT_{loss} = \inf_{\theta\in\mathbb{R}}\int_0^1 |cdf_u^{-1}(q) - (cdf_v-\theta)^{-1}(q)|^p\ \mathrm{d}q\]

For p=1, [45]

\[W_1(u,v) = \int_0^1 |F_u(t)-F_v(t)-LevMed(F_u-F_v)|\ \mathrm{d}t\]

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

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

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

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

Parameters:

• u_values (ndarray, shape (n, ...)) – samples in the source domain (coordinates on [0,1[)

• v_values (ndarray, shape (n, ...)) – samples in the target domain (coordinates on [0,1[)

• u_weights (ndarray, shape (n, ...), optional) – samples weights in the source domain

• v_weights (ndarray, shape (n, ...), optional) – samples weights in the target domain

• p (float, optional (default=1)) – Power p used for computing the Wasserstein distance

• Lm (int, optional) – Lower bound dC. For p>1.

• Lp (int, optional) – Upper bound dC. For p>1.

• tm (float, optional) – Lower bound theta. For p>1.

• tp (float, optional) – Upper bound theta. For p>1.

• eps (float, optional) – Stopping condition. For p>1.

• require_sort (bool, optional) – If True, sort the values.

Returns:

loss – Batched cost associated to the optimal transportation

Return type:

float/array-like, shape (…)

Examples

>>> u = np.array([[0.2,0.5,0.8]])%1
>>> v = np.array([[0.4,0.5,0.7]])%1
>>> wasserstein_circle(u.T, v.T)
array([0.1])

References