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<!DOCTYPE html>

<head>

<title>ot.lowrank — POT Python Optimal Transport 0.9.7 documentation</title>

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

"""

Low rank OT solvers

"""

# Author: Laurène David <laurene.david@ip-paris.fr>

# Titouan Vayer <titouan.vayer@inria.fr>

#

# License: MIT License

import warnings

from .utils import unif, dist, get_lowrank_lazytensor

from .backend import get_backend

from .bregman import sinkhorn

# test if sklearn is installed for linux-minimal-deps

try:

import sklearn.cluster

sklearn_import = True

except ImportError:

sklearn_import = False

def _init_lr_sinkhorn(X_s, X_t, a, b, rank, init, reg_init, random_state, nx=None):

"""

Implementation of different initialization strategies for the low rank sinkhorn solver (Q ,R, g).

This function is specific to lowrank_sinkhorn.

Parameters

----------

X_s : array-like, shape (n_samples_a, dim)

samples in the source domain

X_t : array-like, shape (n_samples_b, dim)

samples in the target domain

a : array-like, shape (n_samples_a,)

samples weights in the source domain

b : array-like, shape (n_samples_b,)

samples weights in the target domain

rank : int

Nonnegative rank of the OT plan.

init : str

Initialization strategy for Q, R and g. 'random', 'trivial' or 'kmeans'

reg_init : float, optional.

Regularization term for a 'kmeans' init.

random_state : int, optional.

Random state for a "random" or 'kmeans' init strategy

nx : optional, Default is None

POT backend

Returns

---------

Q : array-like, shape (n_samples_a, r)

Init for the first low-rank matrix decomposition of the OT plan (Q)

R: array-like, shape (n_samples_b, r)

Init for the second low-rank matrix decomposition of the OT plan (R)

g : array-like, shape (r, )

Init for the weight vector of the low-rank decomposition of the OT plan (g)

References

-----------

.. [65] Scetbon, M., Cuturi, M., & Peyré, G. (2021).

"Low-rank Sinkhorn factorization". In International Conference on Machine Learning.

"""

if nx is None:

nx = get_backend(X_s, X_t, a, b)

ns = X_s.shape[0]

nt = X_t.shape[0]

r = rank

if init == "random":

nx.seed(seed=random_state)

# Init g

g = nx.abs(nx.randn(r, type_as=X_s)) + 1

g = g / nx.sum(g)

# Init Q

Q = nx.abs(nx.randn(ns, r, type_as=X_s)) + 1

Q = (Q.T * (a / nx.sum(Q, axis=1))).T

# Init R

R = nx.abs(nx.randn(nt, rank, type_as=X_s)) + 1

R = (R.T * (b / nx.sum(R, axis=1))).T

if init == "deterministic":

# Init g

g = nx.ones(rank) / rank

lambda_1 = min(nx.min(a), nx.min(g), nx.min(b)) / 2

a1 = nx.arange(start=1, stop=ns + 1, type_as=X_s)

a1 = a1 / nx.sum(a1)

a2 = (a - lambda_1 * a1) / (1 - lambda_1)

b1 = nx.arange(start=1, stop=nt + 1, type_as=X_s)

b1 = b1 / nx.sum(b1)

b2 = (b - lambda_1 * b1) / (1 - lambda_1)

g1 = nx.arange(start=1, stop=rank + 1, type_as=X_s)

g1 = g1 / nx.sum(g1)

g2 = (g - lambda_1 * g1) / (1 - lambda_1)

# Init Q

Q1 = lambda_1 * nx.dot(a1[:, None], nx.reshape(g1, (1, -1)))

Q2 = (1 - lambda_1) * nx.dot(a2[:, None], nx.reshape(g2, (1, -1)))

Q = Q1 + Q2

# Init R

R1 = lambda_1 * nx.dot(b1[:, None], nx.reshape(g1, (1, -1)))

R2 = (1 - lambda_1) * nx.dot(b2[:, None], nx.reshape(g2, (1, -1)))

R = R1 + R2

if init == "kmeans":

if sklearn_import:

# Init g

g = nx.ones(rank, type_as=X_s) / rank

# Init Q

kmeans_Xs = sklearn.cluster.KMeans(

n_clusters=rank, random_state=random_state, n_init="auto"

)

kmeans_Xs.fit(X_s)

Z_Xs = nx.from_numpy(kmeans_Xs.cluster_centers_)

C_Xs = dist(X_s, Z_Xs) # shape (ns, rank)

C_Xs = C_Xs / nx.max(C_Xs)

Q = sinkhorn(a, g, C_Xs, reg=reg_init, numItermax=10000, stopThr=1e-3)

# Init R

kmeans_Xt = sklearn.cluster.KMeans(

)

kmeans_Xt.fit(X_t)

Z_Xt = nx.from_numpy(kmeans_Xt.cluster_centers_)

C_Xt = dist(X_t, Z_Xt) # shape (nt, rank)

C_Xt = C_Xt / nx.max(C_Xt)

R = sinkhorn(b, g, C_Xt, reg=reg_init, numItermax=10000, stopThr=1e-3)

else:

raise ImportError(

"Scikit-learn should be installed to use the 'kmeans' init."

)

return Q, R, g

def compute_lr_sqeuclidean_matrix(X_s, X_t, rescale_cost, nx=None):

"""

Compute the low rank decomposition of a squared euclidean distance matrix.

This function won't work for other distance metrics.

See "Section 3.5, proposition 1"

Parameters

----------

X_s : array-like, shape (n_samples_a, dim)

samples in the source domain

X_t : array-like, shape (n_samples_b, dim)

samples in the target domain

rescale_cost : bool

Rescale the low rank factorization of the sqeuclidean cost matrix

nx : default None

POT backend

Returns

----------

M1 : array-like, shape (n_samples_a, dim+2)

First low rank decomposition of the distance matrix

M2 : array-like, shape (n_samples_b, dim+2)

Second low rank decomposition of the distance matrix

References

-----------

.. [65] Scetbon, M., Cuturi, M., & Peyré, G. (2021).

"Low-rank Sinkhorn factorization". In International Conference on Machine Learning.

"""

if nx is None:

# First low rank decomposition of the cost matrix (A)

array1 = nx.reshape(nx.sum(X_s**2, 1), (-1, 1))

array2 = nx.ones((ns, 1), type_as=X_s)

M1 = nx.concatenate((array1, array2, -2 * X_s), axis=1)

# Second low rank decomposition of the cost matrix (B)

array1 = nx.ones((nt, 1), type_as=X_s)

array2 = nx.reshape(nx.sum(X_t**2, 1), (-1, 1))

M2 = nx.concatenate((array1, array2, X_t), axis=1)

if rescale_cost is True:

M1 = M1 / nx.sqrt(nx.max(M1))

M2 = M2 / nx.sqrt(nx.max(M2))

return M1, M2</div>

def _LR_Dysktra(eps1, eps2, eps3, p1, p2, alpha, stopThr, numItermax, warn, nx=None):

"""

Implementation of the Dykstra algorithm for the Low Rank sinkhorn OT solver.

This function is specific to lowrank_sinkhorn.

Parameters

----------

eps1 : array-like, shape (n_samples_a, r)

First input parameter of the Dykstra algorithm

eps2 : array-like, shape (n_samples_b, r)

Second input parameter of the Dykstra algorithm

eps3 : array-like, shape (r,)

Third input parameter of the Dykstra algorithm

p1 : array-like, shape (n_samples_a,)

Samples weights in the source domain (same as "a" in lowrank_sinkhorn)

p2 : array-like, shape (n_samples_b,)

Samples weights in the target domain (same as "b" in lowrank_sinkhorn)

alpha: int

Lower bound for the weight vector g (same as "alpha" in lowrank_sinkhorn)

stopThr : float

Stop threshold on error

numItermax : int

Max number of iterations

warn : bool, optional

if True, raises a warning if the algorithm doesn't convergence.

nx : default None

POT backend

Returns

----------

Q : array-like, shape (n_samples_a, r)

Dykstra update of the first low-rank matrix decomposition Q

R: array-like, shape (n_samples_b, r)

Dykstra update of the Second low-rank matrix decomposition R

g : array-like, shape (r, )

Dykstra update of the weight vector g

References

----------

.. [65] Scetbon, M., Cuturi, M., & Peyré, G. (2021).

"Low-rank Sinkhorn Factorization". In International Conference on Machine Learning.

"""

# POT backend if None

if nx is None:

nx = get_backend(eps1, eps2, eps3, p1, p2)

# ----------------- Initialisation of Dykstra algorithm -----------------

r = len(eps3) # rank

g_ = nx.copy(eps3) # \tilde{g}

q3_1, q3_2 = nx.ones(r, type_as=p1), nx.ones(r, type_as=p1) # q^{(3)}_1, q^{(3)}_2

v1_, v2_ = (

nx.ones(r, type_as=p1),

) # \tilde{v}^{(1)}, \tilde{v}^{(2)}

q1, q2 = nx.ones(r, type_as=p1), nx.ones(r, type_as=p1) # q^{(1)}, q^{(2)}

err = 1 # initial error

# --------------------- Dykstra algorithm -------------------------

# See Section 3.3 - "Algorithm 2 LR-Dykstra" in paper

for ii in range(numItermax):

if err > stopThr:

# Compute u^{(1)} and u^{(2)}

u1 = p1 / nx.dot(eps1, v1_)

u2 = p2 / nx.dot(eps2, v2_)

# Compute g, g^{(3)}_1 and update \tilde{g}

g = nx.maximum(alpha, g_ * q3_1)

q3_1 = (g_ * q3_1) / g

g_ = nx.copy(g)

# Compute new value of g with \prod

prod1 = (v1_ * q1) * nx.dot(eps1.T, u1)

prod2 = (v2_ * q2) * nx.dot(eps2.T, u2)

g = (g_ * q3_2 * prod1 * prod2) ** (1 / 3)

# Compute v^{(1)} and v^{(2)}

v1 = g / nx.dot(eps1.T, u1)

v2 = g / nx.dot(eps2.T, u2)

# Compute q^{(1)}, q^{(2)} and q^{(3)}_2

q1 = (v1_ * q1) / v1

q2 = (v2_ * q2) / v2

q3_2 = (g_ * q3_2) / g

# Update values of \tilde{v}^{(1)}, \tilde{v}^{(2)} and \tilde{g}

v1_, v2_ = nx.copy(v1), nx.copy(v2)

g_ = nx.copy(g)

# Compute error

err1 = nx.sum(nx.abs(u1 * (eps1 @ v1) - p1))

err2 = nx.sum(nx.abs(u2 * (eps2 @ v2) - p2))

err = err1 + err2

else:

break

else:

if warn:

warnings.warn(

"Dykstra did not converge. You might want to "

"increase the number of iterations `numItermax` "

)

# Compute low rank matrices Q, R

Q = u1[:, None] * eps1 * v1[None, :]

R = u2[:, None] * eps2 * v2[None, :]

return Q, R, g

def lowrank_sinkhorn(

X_s,

X_t,

a=None,

b=None,

reg=0,

rank=None,

alpha=1e-10,

rescale_cost=True,

init="random",

reg_init=1e-1,

seed_init=49,

gamma_init="rescale",

numItermax=2000,

stopThr=1e-7,

warn=True,

log=False,

):

r"""

Solve the entropic regularization optimal transport problem under low-nonnegative rank constraints

on the couplings.

The function solves the following optimization problem:

.. math::

\mathop{\inf_{(\mathbf{Q},\mathbf{R},\mathbf{g}) \in \mathcal{C}(\mathbf{a},\mathbf{b},r)}} \langle \mathbf{C}, \mathbf{Q}\mathrm{diag}(1/\mathbf{g})\mathbf{R}^\top \rangle -

\mathrm{reg} \cdot H((\mathbf{Q}, \mathbf{R}, \mathbf{g}))

where :

- :math:`\mathbf{C}` is the (`dim_a`, `dim_b`) metric cost matrix

- :math:`H((\mathbf{Q}, \mathbf{R}, \mathbf{g}))` is the values of the three respective entropies evaluated for each term.

- :math:`\mathbf{Q}` and :math:`\mathbf{R}` are the low-rank matrix decomposition of the OT plan

- :math:`\mathbf{g}` is the weight vector for the low-rank decomposition of the OT plan

- :math:`\mathbf{a}` and :math:`\mathbf{b}` are source and target weights (histograms, both sum to 1)

- :math:`r` is the rank of the OT plan

- :math:`\mathcal{C}(\mathbf{a}, \mathbf{b}, r)` are the low-rank couplings of the OT problem

Parameters

----------

X_s : array-like, shape (n_samples_a, dim)

samples in the source domain

X_t : array-like, shape (n_samples_b, dim)

samples in the target domain

a : array-like, shape (n_samples_a,)

samples weights in the source domain

b : array-like, shape (n_samples_b,)

samples weights in the target domain

reg : float, optional

Regularization term >0

rank : int, optional. Default is None. (>0)

Nonnegative rank of the OT plan. If None, min(ns, nt) is considered.

alpha : int, optional. Default is 1e-10. (>0 and <1/r)

Lower bound for the weight vector g.

rescale_cost : bool, optional. Default is True

Rescale the low rank factorization of the sqeuclidean cost matrix

init : str, optional. Default is 'random'.

Initialization strategy for the low rank couplings. 'random', 'deterministic' or 'kmeans'

reg_init : float, optional. Default is 1e-1. (>0)

Regularization term for a 'kmeans' init. If None, 1 is considered.

seed_init : int, optional. Default is 49. (>0)

Random state for a 'random' or 'kmeans' init strategy.

gamma_init : str, optional. Default is "rescale".

Initialization strategy for gamma. 'rescale', or 'theory'

Gamma is a constant that scales the convergence criterion of the Mirror Descent

optimization scheme used to compute the low-rank couplings (Q, R and g)

numItermax : int, optional. Default is 2000.

Max number of iterations for the Dykstra algorithm

stopThr : float, optional. Default is 1e-7.

Stop threshold on error (>0) in Dykstra

warn : bool, optional

if True, raises a warning if the algorithm doesn't convergence.

log : bool, optional

record log if True

Returns

---------

Q : array-like, shape (n_samples_a, r)

First low-rank matrix decomposition of the OT plan

R: array-like, shape (n_samples_b, r)

Second low-rank matrix decomposition of the OT plan

g : array-like, shape (r, )

Weight vector for the low-rank decomposition of the OT

log : dict (lazy_plan, value and value_linear)

log dictionary return only if log==True in parameters

References

----------

.. [65] Scetbon, M., Cuturi, M., & Peyré, G. (2021).

"Low-rank Sinkhorn Factorization". In International Conference on Machine Learning.

"""

# POT backend

ns, nt = X_s.shape[0], X_t.shape[0]

# Initialize weights a, b

if a is None:

a = unif(ns, type_as=X_s)

if b is None:

b = unif(nt, type_as=X_t)

# Compute rank (see Section 3.1, def 1)

r = rank

if rank is None:

r = min(ns, nt)

else:

if r <= 0:

raise ValueError("The rank parameter cannot have a negative value")

# Dykstra won't converge if 1/rank < alpha (see Section 3.2)

if 1 / r < alpha:

raise ValueError(

"alpha ({a}) should be smaller than 1/rank ({r}) for the Dykstra algorithm to converge.".format(

a=alpha, r=1 / rank

)

# Low rank decomposition of the sqeuclidean cost matrix

M1, M2 = compute_lr_sqeuclidean_matrix(X_s, X_t, rescale_cost, nx)

# Initialize the low rank matrices Q, R, g

Q, R, g = _init_lr_sinkhorn(X_s, X_t, a, b, r, init, reg_init, seed_init, nx=nx)

# Gamma initialization

if gamma_init == "theory":

L = nx.sqrt(

3 * (2 / (alpha**4)) * ((nx.norm(M1) * nx.norm(M2)) ** 2)

+ (reg + (2 / (alpha**3)) * (nx.norm(M1) * nx.norm(M2))) ** 2

)

gamma = 1 / (2 * L)

if gamma_init not in ["rescale", "theory"]:

raise (

NotImplementedError('Not implemented gamma_init="{}"'.format(gamma_init))

)

# -------------------------- Low rank algorithm ------------------------------

# see "Section 3.3, Algorithm 3 LOT"

for ii in range(100):

# Compute C*R dot using the lr decomposition of C

CR = nx.dot(M2.T, R)

CR_ = nx.dot(M1, CR)

diag_g = (1 / g)[None, :]

CR_g = CR_ * diag_g

# Compute C.T * Q using the lr decomposition of C

CQ = nx.dot(M1.T, Q)

CQ_ = nx.dot(M2, CQ)

CQ_g = CQ_ * diag_g

# Compute omega

omega = nx.diag(nx.dot(Q.T, CR_))

# Rescale gamma at each iteration

if gamma_init == "rescale":

norm_1 = nx.max(nx.abs(CR_ * diag_g + reg * nx.log(Q))) ** 2

norm_2 = nx.max(nx.abs(CQ_ * diag_g + reg * nx.log(R))) ** 2

norm_3 = nx.max(nx.abs(-omega * diag_g)) ** 2

gamma = 10 / max(norm_1, norm_2, norm_3)

eps1 = nx.exp(-gamma * CR_g - ((gamma * reg) - 1) * nx.log(Q))

eps2 = nx.exp(-gamma * CQ_g - ((gamma * reg) - 1) * nx.log(R))

eps3 = nx.exp((gamma * omega / (g**2)) - (gamma * reg - 1) * nx.log(g))

# LR Dykstra algorithm

eps1, eps2, eps3, a, b, alpha, stopThr, numItermax, warn, nx

)

Q = Q + 1e-16

R = R + 1e-16

g = g + 1e-16

# ----------------- Compute lazy_plan, value and value_linear ------------------

# see "Section 3.2: The Low-rank OT Problem" in the paper

# Compute lazy plan (using LazyTensor class)

lazy_plan = get_lowrank_lazytensor(Q, R, 1 / g)

# Compute value_linear (using trace formula)

A = nx.dot(M1.T, Q)

B = nx.dot(M2.T, R * diag_g)

value_linear = nx.sum(A * B)

# Compute value with entropy reg (see "Section 3.2" in the paper)

reg_Q = nx.sum(Q * nx.log(Q + 1e-16)) # entropy for Q

reg_g = nx.sum(g * nx.log(g + 1e-16)) # entropy for g

reg_R = nx.sum(R * nx.log(R + 1e-16)) # entropy for R

value = value_linear + reg * (reg_Q + reg_g + reg_R)

if log:

dict_log = dict()

dict_log["value"] = value

dict_log["value_linear"] = value_linear

dict_log["lazy_plan"] = lazy_plan

return Q, R, g, dict_log

return Q, R, g</div>

def kernel_nystroem(X_s, X_t, anchors=50, sigma=1.0, random_state=None):

r"""

Compute left and right factors corresponding to the Nystroem method on the Gaussian kernel :math:`K(x^s_i, x^t_j) = \exp(-\|x^s_i-x^t_j\|^2/2\sigma^2)`.

The Nystroem approximation is computed by sampling :math:`\min(n, \lceil(c / 2))\rceil)` components in each distribution,

where :math:`n` is the number of samples in the distribution and :math:`c` the total number of anchor points.

Parameters

----------

X_s : array-like, shape (n_samples_a, dim)

samples in the source domain

X_t : array-like, shape (n_samples_b, dim)

samples in the target domain

anchors : int, optional

The total number of anchors sampled for the Nystroem approximation (anchors/2 in each distribution), default 50.

sigma : float, optional

The standard deviation parameter for the Gaussian kernel.

random_state : int, optional

The random state for sampling the components in each distribution.

Returns

-------

left_factor : array-like, shape (n_samples_a, dim_r)

Left factor of Nystroem

right_factor : array-like, shape (n_samples_b, dim_r)

Right factor of Nystroem

"""

if random_state is not None:

nx.seed(random_state)

n, m = X_s.shape[0], X_t.shape[0]

if not isinstance(anchors, int) or anchors < 2:

raise ValueError("anchors must be an integer >= 2")

n_components_source = min(n, anchors // 2)

n_components_target = min(m, anchors // 2)

# draw n_components/2 points in each distribution

inds_source = nx.randperm(n) # sample n_components_source uniformly

basis_source = X_s[inds_source[:n_components_source]]

inds_target = nx.randperm(m)

basis_target = X_t[inds_target[:n_components_target]]

basis = nx.concatenate((basis_source, basis_target))

Mzz = dist(basis, metric="sqeuclidean") # compute \|z_i - z_j\|_2^2

basis_kernel = nx.exp(-Mzz / (2.0 * sigma**2))

normalization = nx.pinv(basis_kernel, hermitian=True)

Mxz = dist(X_s, basis, metric="sqeuclidean")

Myz = dist(X_t, basis, metric="sqeuclidean")

left_factor = nx.exp(-Mxz / (2.0 * sigma**2)) @ normalization

right_factor = nx.exp(-Myz / (2.0 * sigma**2))

return left_factor, right_factor # left_factor @ right_factor.T approx K</div>

def sinkhorn_low_rank_kernel(

K1, # left factor

K2, # right factor

a=None,

b=None,

numItermax=1000,

stopThr=1e-9,

verbose=False,

log=False,

warn=True,

warmstart=None,

):

r"""

Compute the Sinkhorn algorithm for a kernel :math:`\mathbf{K}` that can be written as a low rank factorization :math:`\mathbf{K} = \mathbf{K}_1 \mathbf{K}_2^\top`.

Does not implement multiple targets.

Precisely :

- :math:`\mathbf{K}_1, \mathbf{K}_2` are the (`dim_a`, `dim_r`), (`dim_b`, `dim_r`) kernel matrices

- :math:`\Omega` is the entropic regularization term

:math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`

- :math:`\mathbf{a}` and :math:`\mathbf{b}` are source and target

weights (histograms, both sum to 1)

The algorithm used for solving the problem is the Sinkhorn-Knopp

matrix scaling algorithm as proposed in :ref:`[2] <references-sinkhorn-knopp>`

Parameters

----------

K_1 : array-like, shape (n_samples_a, dim_r)

Left factor

K_2 : array-like, shape (n_samples_b, dim_r)

Right factor

a : array-like, shape (n_samples_a,)

samples weights in the source domain

b : array-like, shape (n_samples_b,)

samples in the target domain

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

warn : bool, optional

if True, raises a warning if the algorithm doesn't convergence.

warmstart: tuple of arrays, shape (n_samples_a, n_samples_b), optional

Initialization of dual potentials. If provided, the dual potentials should be given

(that is the logarithm of the u,v sinkhorn scaling vectors)

Returns

---------

u : array-like, shape (n_samples_a, )

Left dual variable

v: array-like, shape (n_samples_b, )

Right dual variable

log : dict (lazy_plan)

log dictionary return only if log==True in parameters

"""

nx = get_backend(K1, K2, a, b)

if a is None:

a = nx.full((K1.shape[0],), 1.0 / K1.shape[0], type_as=K1)

if b is None:

b = nx.full((K2.shape[0],), 1.0 / K2.shape[0], type_as=K2)

# init data

dim_a = len(a)

dim_b = b.shape[0]

if log:

dict_log = {"err": []}

# we assume that no distances are null except those of the diagonal of

# distances

if warmstart is None:

u = nx.ones(dim_a, type_as=K1) / dim_a

v = nx.ones(dim_b, type_as=K2) / dim_b

else:

u, v = nx.exp(warmstart[0]), nx.exp(warmstart[1])

err = 1

for ii in range(numItermax):

uprev = u

vprev = v

KtransposeU = K2 @ (nx.transpose(K1) @ u)

v = b / KtransposeU

KV = K1 @ (nx.transpose(K2) @ v)

u = a / KV

if (

nx.any(KtransposeU == 0)

or nx.any(nx.isnan(u))

or nx.any(nx.isinf(u))

or nx.any(nx.isinf(v))

):

# we have reached the machine precision

# come back to previous solution and quit loop

warnings.warn("Warning: numerical errors at iteration %d" % ii)

u = uprev

v = vprev

break

if ii % 10 == 0:

# we can speed up the process by checking for the error only all

# the 10th iterations

tmp2 = (

v * (K2 @ (nx.transpose(K1) @ u))

) # same as nx.einsum("ik, ir, jr, jk->jk", u, K1, K2, v) and nx.einsum("i, ir, jr, j->j", u, K1, K2, v) in the scalar case

err = nx.norm(tmp2 - b) # violation of marginal

if log:

dict_log["err"].append(err)

if err < stopThr:

break

if verbose:

if ii % 200 == 0:

print("{:5s}|{:12s}".format("It.", "Err") + "\n" + "-" * 19)

print("{:5d}|{:8e}|".format(ii, err))

else:

if warn:

warnings.warn(

"Sinkhorn did not converge. You might want to "

"increase the number of iterations `numItermax` "

"or the regularization parameter `reg`."

)

if log:

dict_log["niter"] = ii

dict_log["u"] = u

dict_log["v"] = v

dict_log["lazy_plan"] = get_lowrank_lazytensor(

u.reshape((-1, 1)) * K1, v.reshape((-1, 1)) * K2

)

return u, v, dict_log

else:

return u, v</div>

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