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<!DOCTYPE html>

<head>

<title>ot.lp — POT Python Optimal Transport 0.9.6dev0 documentation</title>

</head>

POT Python Optimal Transport

</a>

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

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<li class="breadcrumb-item"><a href="../index.html">Module code</a></li>

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</div>

<h1>Source code for ot.lp</h1><div class="highlight"><pre>

# -*- coding: utf-8 -*-

"""

Solvers for the original linear program OT problem.

"""

# Author: Remi Flamary <remi.flamary@unice.fr>

#

# License: MIT License

import numpy as np

import warnings

from . import cvx

from .cvx import barycenter

from .dmmot import dmmot_monge_1dgrid_loss, dmmot_monge_1dgrid_optimize

# import compiled emd

from .emd_wrap import emd_c, check_result, emd_1d_sorted

from .solver_1d import (

emd_1d,

emd2_1d,

wasserstein_1d,

binary_search_circle,

wasserstein_circle,

semidiscrete_wasserstein2_unif_circle,

)

from ..utils import dist, list_to_array

from ..backend import get_backend

__all__ = [

"emd",

"emd2",

"barycenter",

"free_support_barycenter",

"cvx",

"emd_1d_sorted",

"emd_1d",

"emd2_1d",

"wasserstein_1d",

"generalized_free_support_barycenter",

"binary_search_circle",

"wasserstein_circle",

"semidiscrete_wasserstein2_unif_circle",

"dmmot_monge_1dgrid_loss",

"dmmot_monge_1dgrid_optimize",

]

def check_number_threads(numThreads):

"""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 : int

Corrected number of threads

"""

if (numThreads is None) or (

isinstance(numThreads, str) and numThreads.lower() == "max"

):

return -1

if (not isinstance(numThreads, int)) or numThreads < 1:

raise ValueError(

'numThreads should either be "max" or a strictly positive integer'

)

return numThreads</div>

def center_ot_dual(alpha0, beta0, a=None, b=None):

r"""Center dual OT potentials w.r.t. their weights

The main idea of this function is to find unique dual potentials

that ensure some kind of centering/fairness. The main idea is to find dual potentials that lead to the same final objective value for both source and targets (see below for more details). It will help having

stability when multiple calling of the OT solver with small changes.

Basically we add another constraint to the potential that will not

change the objective value but will ensure unicity. The constraint

is the following:

.. math::

\alpha^T \mathbf{a} = \beta^T \mathbf{b}

in addition to the OT problem constraints.

since :math:`\sum_i a_i=\sum_j b_j` this can be solved by adding/removing

a constant from both :math:`\alpha_0` and :math:`\beta_0`.

.. math::

c &= \frac{\beta_0^T \mathbf{b} - \alpha_0^T \mathbf{a}}{\mathbf{1}^T \mathbf{b} + \mathbf{1}^T \mathbf{a}}

\alpha &= \alpha_0 + c

\beta &= \beta_0 + c

Parameters

----------

alpha0 : (ns,) numpy.ndarray, float64

Source dual potential

beta0 : (nt,) numpy.ndarray, float64

Target dual potential

a : (ns,) numpy.ndarray, float64

Source histogram (uniform weight if empty list)

b : (nt,) numpy.ndarray, float64

Target histogram (uniform weight if empty list)

Returns

-------

alpha : (ns,) numpy.ndarray, float64

Source centered dual potential

beta : (nt,) numpy.ndarray, float64

Target centered dual potential

"""

# if no weights are provided, use uniform

if a is None:

a = np.ones(alpha0.shape[0]) / alpha0.shape[0]

if b is None:

b = np.ones(beta0.shape[0]) / beta0.shape[0]

# compute constant that balances the weighted sums of the duals

c = (b.dot(beta0) - a.dot(alpha0)) / (a.sum() + b.sum())

# update duals

alpha = alpha0 + c

beta = beta0 - c

return alpha, beta</div>

def estimate_dual_null_weights(alpha0, beta0, a, b, M):

r"""Estimate feasible values for 0-weighted dual potentials

The feasible values are computed efficiently but rather coarsely.

.. warning::

This function is necessary because the C++ solver in `emd_c`

discards all samples in the distributions with

zeros weights. This means that while the primal variable (transport

matrix) is exact, the solver only returns feasible dual potentials

on the samples with weights different from zero.

First we compute the constraints violations:

.. math::

\mathbf{V} = \alpha + \beta^T - \mathbf{M}

Next we compute the max amount of violation per row (:math:`\alpha`) and

columns (:math:`beta`)

.. math::

\mathbf{v^a}_i = \max_j \mathbf{V}_{i,j}

\mathbf{v^b}_j = \max_i \mathbf{V}_{i,j}

Finally we update the dual potential with 0 weights if a

constraint is violated

.. math::

\alpha_i = \alpha_i - \mathbf{v^a}_i \quad \text{ if } \mathbf{a}_i=0 \text{ and } \mathbf{v^a}_i>0

\beta_j = \beta_j - \mathbf{v^b}_j \quad \text{ if } \mathbf{b}_j=0 \text{ and } \mathbf{v^b}_j > 0

In the end the dual potentials are centered using function

:py:func:`ot.lp.center_ot_dual`.

Note that all those updates do not change the objective value of the

solution but provide dual potentials that do not violate the constraints.

Parameters

----------

alpha0 : (ns,) numpy.ndarray, float64

Source dual potential

beta0 : (nt,) numpy.ndarray, float64

Target dual potential

alpha0 : (ns,) numpy.ndarray, float64

Source dual potential

beta0 : (nt,) numpy.ndarray, float64

Target dual potential

a : (ns,) numpy.ndarray, float64

Source distribution (uniform weights if empty list)

b : (nt,) numpy.ndarray, float64

Target distribution (uniform weights if empty list)

M : (ns,nt) numpy.ndarray, float64

Loss matrix (c-order array with type float64)

Returns

-------

alpha : (ns,) numpy.ndarray, float64

Source corrected dual potential

beta : (nt,) numpy.ndarray, float64

Target corrected dual potential

"""

# binary indexing of non-zeros weights

asel = a != 0

bsel = b != 0

# compute dual constraints violation

constraint_violation = alpha0[:, None] + beta0[None, :] - M

# Compute largest violation per line and columns

aviol = np.max(constraint_violation, 1)

bviol = np.max(constraint_violation, 0)

# update corrects violation of

alpha_up = -1 * ~asel * np.maximum(aviol, 0)

beta_up = -1 * ~bsel * np.maximum(bviol, 0)

alpha = alpha0 + alpha_up

beta = beta0 + beta_up

return center_ot_dual(alpha, beta, a, b)</div>

def emd(

a,

b,

M,

numItermax=100000,

log=False,

center_dual=True,

numThreads=1,

check_marginals=True,

):

r"""Solves the Earth Movers distance problem and returns the OT matrix

.. math::

\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

where :

- :math:`\mathbf{M}` is the metric cost matrix

- :math:`\mathbf{a}` and :math:`\mathbf{b}` are the sample weights

.. warning:: Note that the :math:`\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: :math:`\mathbf{a}`,

then :math:`\mathbf{b}`, then :math:`\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 :math:`\mathbf{a}` and :math:`\mathbf{b}` do not sum to the same value.

Uses the algorithm proposed in :ref:`[1] <references-emd>`.

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

:py:func:`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-emd:

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.

See Also

--------

ot.bregman.sinkhorn : Entropic regularized OT

ot.optim.cg : General regularized OT

"""

a, b, M = list_to_array(a, b, M)

nx = get_backend(M, a, b)

if len(a) != 0:

type_as = a

elif len(b) != 0:

type_as = b

else:

type_as = M

# if empty array given then use uniform distributions

if len(a) == 0:

a = nx.ones((M.shape[0],), type_as=type_as) / M.shape[0]

if len(b) == 0:

b = nx.ones((M.shape[1],), type_as=type_as) / M.shape[1]

# convert to numpy

M, a, b = nx.to_numpy(M, a, b)

# ensure float64

a = np.asarray(a, dtype=np.float64)

b = np.asarray(b, dtype=np.float64)

M = np.asarray(M, dtype=np.float64, order="C")

# if empty array given then use uniform distributions

a = np.ones((M.shape[0],), dtype=np.float64) / M.shape[0]

b = np.ones((M.shape[1],), dtype=np.float64) / M.shape[1]

assert (

a.shape[0] == M.shape[0] and b.shape[0] == M.shape[1]

), "Dimension mismatch, check dimensions of M with a and b"

# ensure that same mass

if check_marginals:

np.testing.assert_almost_equal(

a.sum(0),

b.sum(0),

err_msg="a and b vector must have the same sum",

decimal=6,

)

asel = a != 0

bsel = b != 0

numThreads = check_number_threads(numThreads)

G, cost, u, v, result_code = emd_c(a, b, M, numItermax, numThreads)

if center_dual:

u, v = center_ot_dual(u, v, a, b)

if np.any(~asel) or np.any(~bsel):

u, v = estimate_dual_null_weights(u, v, a, b, M)

result_code_string = check_result(result_code)

if not nx.is_floating_point(type_as):

warnings.warn(

"Input histogram consists of integer. The transport plan will be "

"casted accordingly, possibly resulting in a loss of precision. "

"If this behaviour is unwanted, please make sure your input "

"histogram consists of floating point elements.",

stacklevel=2,

)

if log:

log = {}

log["cost"] = cost

log["u"] = nx.from_numpy(u, type_as=type_as)

log["v"] = nx.from_numpy(v, type_as=type_as)

log["warning"] = result_code_string

log["result_code"] = result_code

return nx.from_numpy(G, type_as=type_as), log

def emd2(

a,

b,

M,

processes=1,

numItermax=100000,

log=False,

return_matrix=False,

center_dual=True,

numThreads=1,

check_marginals=True,

):

r"""Solves the Earth Movers distance problem and returns the loss

.. math::

\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

where :

- :math:`\mathbf{M}` is the metric cost matrix

- :math:`\mathbf{a}` and :math:`\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: :math:`\mathbf{a}`, then :math:`\mathbf{b}`,

then :math:`\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 :math:`\mathbf{a}` and :math:`\mathbf{b}` do not sum to the same value.

Uses the algorithm proposed in :ref:`[1] <references-emd2>`.

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

:py:func:`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-emd2:

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.

See Also

--------

ot.bregman.sinkhorn : Entropic regularized OT

ot.optim.cg : General regularized OT

"""

type_as = a

type_as = b

else:

type_as = M

# if empty array given then use uniform distributions

# store original tensors

a0, b0, M0 = a, b, M

# convert to numpy

assert (

), "Dimension mismatch, check dimensions of M with a and b"

# ensure that same mass

if check_marginals:

np.testing.assert_almost_equal(

a.sum(0),

b.sum(0, keepdims=True),

err_msg="a and b vector must have the same sum",

decimal=6,

)

b = b * a.sum(0) / b.sum(0, keepdims=True)

asel = a != 0

numThreads = check_number_threads(numThreads)

if log or return_matrix:

def f(b):

bsel = b != 0

if center_dual:

result_code_string = check_result(result_code)

log = {}

warnings.warn(

"Input histogram consists of integer. The transport plan will be "

"casted accordingly, possibly resulting in a loss of precision. "

"If this behaviour is unwanted, please make sure your input "

"histogram consists of floating point elements.",

stacklevel=2,

)

G = nx.from_numpy(G, type_as=type_as)

if return_matrix:

log["G"] = G

log["warning"] = result_code_string

log["result_code"] = result_code

cost = nx.set_gradients(

nx.from_numpy(cost, type_as=type_as),

(a0, b0, M0),

(log["u"] - nx.mean(log["u"]), log["v"] - nx.mean(log["v"]), G),

)

return [cost, log]

else:

def f(b):

bsel = b != 0

if center_dual:

warnings.warn(

"Input histogram consists of integer. The transport plan will be "

"casted accordingly, possibly resulting in a loss of precision. "

"If this behaviour is unwanted, please make sure your input "

"histogram consists of floating point elements.",

stacklevel=2,

)

cost = nx.set_gradients(

(a0, b0, M0),

(

nx.from_numpy(u - np.mean(u), type_as=type_as),

nx.from_numpy(v - np.mean(v), type_as=type_as),

G,

),

)

check_result(result_code)

return cost

if len(b.shape) == 1:

return f(b)

nb = b.shape[1]

if processes > 1:

warnings.warn(

"The 'processes' parameter has been deprecated. "

"Multiprocessing should be done outside of POT."

)

res = list(map(f, [b[:, i].copy() for i in range(nb)]))

return res</div>

def free_support_barycenter(

measures_locations,

measures_weights,

X_init,

b=None,

weights=None,

numItermax=100,

stopThr=1e-7,

verbose=False,

log=None,

numThreads=1,

):

r"""

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:

.. math::

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

where :

- :math:`w \in \mathbb{(0, 1)}^{N}`'s are the barycenter weights and sum to one

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

- `measures_locations` denotes the :math:`\mathbf{X}_i \in \mathbb{R}^{k_i, d}`: empirical measures atoms locations

- :math:`\mathbf{b} \in \mathbb{R}^{k}` is the desired weights vector of the barycenter

This problem is considered in :ref:`[20] <references-free-support-barycenter>` (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. :math:`\theta = 1` in

:ref:`[20] <references-free-support-barycenter>` (Algorithm 2). This can be seen as a discrete

implementation of the fixed-point algorithm of

:ref:`[43] <references-free-support-barycenter>` proposed in the continuous setting.

Parameters

----------

measures_locations : list of N (k_i,d) array-like

The discrete support of a measure supported on :math:`k_i` locations of a `d`-dimensional space

(:math:`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 :math:`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

----------

.. [20] Cuturi, Marco, and Arnaud Doucet. "Fast computation of Wasserstein barycenters." International Conference on Machine Learning. 2014.

.. [43] Álvarez-Esteban, Pedro C., et al. "A fixed-point approach to barycenters in Wasserstein space." Journal of Mathematical Analysis and Applications 441.2 (2016): 744-762.

"""

nx = get_backend(*measures_locations, *measures_weights, X_init)

iter_count = 0

N = len(measures_locations)

k = X_init.shape[0]

d = X_init.shape[1]

if b is None:

if weights is None:

weights = nx.ones((N,), type_as=X_init) / N

X = X_init

log_dict = {}

displacement_square_norms = []

displacement_square_norm = stopThr + 1.0

while displacement_square_norm > stopThr and iter_count < numItermax:

T_sum = nx.zeros((k, d), type_as=X_init)

for measure_locations_i, measure_weights_i, weight_i in zip(

measures_locations, measures_weights, weights

):

M_i = dist(X, measure_locations_i)

T_i = emd(b, measure_weights_i, M_i, numThreads=numThreads)

T_sum = T_sum + weight_i * 1.0 / b[:, None] * nx.dot(

T_i, measure_locations_i

)

displacement_square_norm = nx.sum((T_sum - X) ** 2)

if log:

displacement_square_norms.append(displacement_square_norm)

X = T_sum

if verbose:

print(

"iteration %d, displacement_square_norm=%f\n",

iter_count,

displacement_square_norm,

)

iter_count += 1

if log:

log_dict["displacement_square_norms"] = displacement_square_norms

return X, log_dict

else:

return X</div>

def generalized_free_support_barycenter(

X_list,

a_list,

P_list,

n_samples_bary,

Y_init=None,

b=None,

weights=None,

numItermax=100,

stopThr=1e-7,

verbose=False,

log=None,

numThreads=1,

eps=0,

):

r"""

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:

.. math::

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

where :

- :math:`\gamma = \sum_{l=1}^n b_l\delta_{y_l}` is the desired barycenter with each :math:`y_l \in \mathbb{R}^d`

- :math:`\mathbf{b} \in \mathbb{R}^{n}` is the desired weights vector of the barycenter

- The input measures are :math:`\nu_i = \sum_{j=1}^{k_i}a_{i,j}\delta_{x_{i,j}}`

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

- The :math:`\mathbf{X}_i \in \mathbb{R}^{k_i, d_i}` are the respective empirical measures atoms locations

- :math:`w = (w_1, \cdots w_p)` are the barycenter coefficients (on the simplex)

- Each :math:`\mathbf{P}_i \in \mathbb{R}^{d, d_i}`, and :math:`P_i\#\nu_i = \sum_{j=1}^{k_i}a_{i,j}\delta_{P_ix_{i,j}}`

As show by :ref:`[42] <references-generalized-free-support-barycenter>`,

this problem can be re-written as a Wasserstein Barycenter problem,

which we solve using the free support method :ref:`[20] <references-generalized-free-support-barycenter>`

(Algorithm 2).

Parameters

----------

X_list : list of p (k_i,d_i) array-like

Discrete supports of the input measures: each consists of :math:`k_i` locations of a `d_i`-dimensional space

(:math:`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 :math:`P_i` is a linear map :math:`\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 :math:`\mathbf{P}_i^T` matrices don't span :math:`\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 : (n_samples_bary,d) array-like

Support locations (on n_samples_bary atoms) of the barycenter

.. _references-generalized-free-support-barycenter:

References

----------

.. [20] Cuturi, M. and Doucet, A.. "Fast computation of Wasserstein barycenters." International Conference on Machine Learning. 2014.

.. [42] Delon, J., Gozlan, N., and Saint-Dizier, A.. Generalized Wasserstein barycenters between probability measures living on different subspaces. arXiv preprint arXiv:2105.09755, 2021.

"""

nx = get_backend(*X_list, *a_list, *P_list)

d = P_list[0].shape[1]

p = len(X_list)

if weights is None:

weights = nx.ones(p, type_as=X_list[0]) / p

# variable change matrix to reduce the problem to a Wasserstein Barycenter (WB)

A = eps * nx.eye(

d, type_as=X_list[0]

) # if eps nonzero: will force the invertibility of A

for P_i, lambda_i in zip(P_list, weights):

A = A + lambda_i * P_i.T @ P_i

B = nx.inv(nx.sqrtm(A))

Z_list = [

x @ Pi @ B.T for (x, Pi) in zip(X_list, P_list)

] # change of variables -> (WB) problem on Z

if Y_init is None:

Y_init = nx.randn(n_samples_bary, d, type_as=X_list[0])

if b is None:

b = nx.ones(n_samples_bary, type_as=X_list[0]) / n_samples_bary # not optimized

out = free_support_barycenter(

Z_list,

a_list,

Y_init,

b,

numItermax=numItermax,

stopThr=stopThr,

verbose=verbose,

log=log,

numThreads=numThreads,

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