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<!DOCTYPE html>

<head>

<title>ot.bregman — POT Python Optimal Transport 0.9.1 documentation</title>

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POT Python Optimal Transport

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0.9.1

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

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

"""

Bregman projections solvers for entropic regularized OT

"""

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

# Nicolas Courty <ncourty@irisa.fr>

# Kilian Fatras <kilian.fatras@irisa.fr>

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

# Hicham Janati <hicham.janati100@gmail.com>

# Mokhtar Z. Alaya <mokhtarzahdi.alaya@gmail.com>

# Alexander Tong <alexander.tong@yale.edu>

# Ievgen Redko <ievgen.redko@univ-st-etienne.fr>

# Quang Huy Tran <quang-huy.tran@univ-ubs.fr>

#

# License: MIT License

import warnings

import numpy as np

from scipy.optimize import fmin_l_bfgs_b

from ot.utils import unif, dist, list_to_array

from .backend import get_backend

<div class="viewcode-block" id="sinkhorn"><a class="viewcode-back" href="../../gen_modules/ot.bregman.html#ot.sinkhorn">[docs]</a>def sinkhorn(a, b, M, reg, method='sinkhorn', numItermax=1000, stopThr=1e-9,

verbose=False, log=False, warn=True, warmstart=None, **kwargs):

r"""

Solve the entropic regularization optimal transport problem and return the OT matrix

The function solves the following optimization problem:

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg}\cdot\Omega(\gamma)

s.t. \ \gamma \mathbf{1} &= \mathbf{a}

\gamma^T \mathbf{1} &= \mathbf{b}

\gamma &\geq 0

where :

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

- :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)

.. note:: This function is backend-compatible and will work on arrays

from all compatible backends.

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

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

**Choosing a Sinkhorn solver**

By default and when using a regularization parameter that is not too small

the default sinkhorn solver should be enough. If you need to use a small

regularization to get sharper OT matrices, you should use the

:py:func:`ot.bregman.sinkhorn_stabilized` solver that will avoid numerical

errors. This last solver can be very slow in practice and might not even

converge to a reasonable OT matrix in a finite time. This is why

:py:func:`ot.bregman.sinkhorn_epsilon_scaling` that relies on iterating the value

of the regularization (and using warm start) sometimes leads to better

solutions. Note that the greedy version of the sinkhorn

:py:func:`ot.bregman.greenkhorn` can also lead to a speedup and the screening

version of the sinkhorn :py:func:`ot.bregman.screenkhorn` aim at providing a

fast approximation of the Sinkhorn problem. For use of GPU and gradient

computation with small number of iterations we strongly recommend the

:py:func:`ot.bregman.sinkhorn_log` solver that will no need to check for

numerical problems.

Parameters

----------

a : array-like, shape (dim_a,)

samples weights in the source domain

b : array-like, shape (dim_b,) or ndarray, shape (dim_b, n_hists)

samples in the target domain, compute sinkhorn with multiple targets

and fixed :math:`\mathbf{M}` if :math:`\mathbf{b}` is a matrix

(return OT loss + dual variables in log)

M : array-like, shape (dim_a, dim_b)

loss matrix

reg : float

Regularization term >0

method : str

method used for the solver either 'sinkhorn','sinkhorn_log',

'greenkhorn', 'sinkhorn_stabilized' or 'sinkhorn_epsilon_scaling', see

those function for specific parameters

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 (dim_a, dim_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

-------

gamma : array-like, shape (dim_a, dim_b)

Optimal transportation matrix for the given parameters

log : dict

log dictionary return only if log==True in parameters

Examples

--------

>>> import ot

>>> a=[.5, .5]

>>> b=[.5, .5]

>>> M=[[0., 1.], [1., 0.]]

>>> ot.sinkhorn(a, b, M, 1)

array([[0.36552929, 0.13447071],

[0.13447071, 0.36552929]])

.. _references-sinkhorn:

References

----------

.. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation

of Optimal Transport, Advances in Neural Information Processing

Systems (NIPS) 26, 2013

.. [9] Schmitzer, B. (2016). Stabilized Sparse Scaling Algorithms

for Entropy Regularized Transport Problems. arXiv preprint arXiv:1610.06519.

.. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016).

Scaling algorithms for unbalanced transport problems.

arXiv preprint arXiv:1607.05816.

.. [34] Feydy, J., Séjourné, T., Vialard, F. X., Amari, S. I., Trouvé,

A., & Peyré, G. (2019, April). Interpolating between optimal transport

and MMD using Sinkhorn divergences. In The 22nd International Conference

on Artificial Intelligence and Statistics (pp. 2681-2690). PMLR.

See Also

--------

ot.lp.emd : Unregularized OT

ot.optim.cg : General regularized OT

ot.bregman.sinkhorn_knopp : Classic Sinkhorn :ref:`[2] <references-sinkhorn>`

ot.bregman.sinkhorn_stabilized: Stabilized sinkhorn

:ref:`[9] <references-sinkhorn>` :ref:`[10] <references-sinkhorn>`

ot.bregman.sinkhorn_epsilon_scaling: Sinkhorn with epsilon scaling

:ref:`[9] <references-sinkhorn>` :ref:`[10] <references-sinkhorn>`

"""

if method.lower() == 'sinkhorn':

return sinkhorn_knopp(a, b, M, reg, numItermax=numItermax,

stopThr=stopThr, verbose=verbose, log=log,

warn=warn, warmstart=warmstart,

**kwargs)

elif method.lower() == 'sinkhorn_log':

return sinkhorn_log(a, b, M, reg, numItermax=numItermax,

**kwargs)

return greenkhorn(a, b, M, reg, numItermax=numItermax,

return sinkhorn_stabilized(a, b, M, reg, numItermax=numItermax,

stopThr=stopThr, warmstart=warmstart,

**kwargs)

return sinkhorn_epsilon_scaling(a, b, M, reg, numItermax=numItermax,

**kwargs)

else:

raise ValueError("Unknown method '%s'." % method)</div>

<div class="viewcode-block" id="sinkhorn2"><a class="viewcode-back" href="../../gen_modules/ot.bregman.html#ot.sinkhorn2">[docs]</a>def sinkhorn2(a, b, M, reg, method='sinkhorn', numItermax=1000,

stopThr=1e-9, verbose=False, log=False, warn=False, warmstart=None, **kwargs):

r"""

Solve the entropic regularization optimal transport problem and return the loss

The function solves the following optimization problem:

.. math::

W = \min_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg}\cdot\Omega(\gamma)

s.t. \ \gamma \mathbf{1} &= \mathbf{a}

\gamma^T \mathbf{1} &= \mathbf{b}

\gamma &\geq 0

where :

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

- :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)

and returns :math:`\langle \gamma^*, \mathbf{M} \rangle_F` (without

the entropic contribution).

.. note:: This function is backend-compatible and will work on arrays

from all compatible backends.

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

scaling algorithm as proposed in :ref:`[2] <references-sinkhorn2>`

**Choosing a Sinkhorn solver**

By default and when using a regularization parameter that is not too small

the default sinkhorn solver should be enough. If you need to use a small

regularization to get sharper OT matrices, you should use the

:py:func:`ot.bregman.sinkhorn_log` solver that will avoid numerical

errors. This last solver can be very slow in practice and might not even

converge to a reasonable OT matrix in a finite time. This is why

:py:func:`ot.bregman.sinkhorn_epsilon_scaling` that relies on iterating the value

of the regularization (and using warm start) sometimes leads to better

solutions. Note that the greedy version of the sinkhorn

:py:func:`ot.bregman.greenkhorn` can also lead to a speedup and the screening

version of the sinkhorn :py:func:`ot.bregman.screenkhorn` aim a providing a

fast approximation of the Sinkhorn problem. For use of GPU and gradient

computation with small number of iterations we strongly recommend the

:py:func:`ot.bregman.sinkhorn_log` solver that will no need to check for

numerical problems.

Parameters

----------

a : array-like, shape (dim_a,)

samples weights in the source domain

b : array-like, shape (dim_b,) or ndarray, shape (dim_b, n_hists)

samples in the target domain, compute sinkhorn with multiple targets

and fixed :math:`\mathbf{M}` if :math:`\mathbf{b}` is a matrix

(return OT loss + dual variables in log)

M : array-like, shape (dim_a, dim_b)

loss matrix

reg : float

Regularization term >0

method : str

method used for the solver either 'sinkhorn','sinkhorn_log',

'sinkhorn_stabilized', see those function for specific parameters

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 (dim_a, dim_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

-------

W : (n_hists) float/array-like

Optimal transportation loss for the given parameters

log : dict

log dictionary return only if log==True in parameters

Examples

--------

>>> import ot

>>> a=[.5, .5]

>>> b=[.5, .5]

>>> M=[[0., 1.], [1., 0.]]

>>> ot.sinkhorn2(a, b, M, 1)

0.26894142136999516

.. _references-sinkhorn2:

References

----------

.. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of

Optimal Transport, Advances in Neural Information

Processing Systems (NIPS) 26, 2013

.. [9] Schmitzer, B. (2016). Stabilized Sparse Scaling Algorithms

for Entropy Regularized Transport Problems.

arXiv preprint arXiv:1610.06519.

.. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016).

Scaling algorithms for unbalanced transport problems.

arXiv preprint arXiv:1607.05816.

.. [21] Altschuler J., Weed J., Rigollet P. : Near-linear time approximation

algorithms for optimal transport via Sinkhorn iteration,

Advances in Neural Information Processing Systems (NIPS) 31, 2017

.. [34] Feydy, J., Séjourné, T., Vialard, F. X., Amari, S. I.,

Trouvé, A., & Peyré, G. (2019, April).

Interpolating between optimal transport and MMD using Sinkhorn

divergences. In The 22nd International Conference on Artificial

Intelligence and Statistics (pp. 2681-2690). PMLR.

See Also

--------

ot.lp.emd : Unregularized OT

ot.optim.cg : General regularized OT

ot.bregman.sinkhorn_knopp : Classic Sinkhorn :ref:`[2] <references-sinkhorn2>`

ot.bregman.greenkhorn : Greenkhorn :ref:`[21] <references-sinkhorn2>`

ot.bregman.sinkhorn_stabilized: Stabilized sinkhorn

:ref:`[9] <references-sinkhorn2>` :ref:`[10] <references-sinkhorn2>`

"""

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

nx = get_backend(M, a, b)

if len(b.shape) < 2:

res = sinkhorn_knopp(a, b, M, reg, numItermax=numItermax,

log=log, warn=warn, warmstart=warmstart,

**kwargs)

res = sinkhorn_log(a, b, M, reg, numItermax=numItermax,

**kwargs)

res = sinkhorn_stabilized(a, b, M, reg, numItermax=numItermax,

**kwargs)

else:

if log:

return nx.sum(M * res[0]), res[1]

else:

**kwargs)

else:

<div class="viewcode-block" id="sinkhorn_knopp"><a class="viewcode-back" href="../../gen_modules/ot.bregman.html#ot.sinkhorn_knopp">[docs]</a>def sinkhorn_knopp(a, b, M, reg, numItermax=1000, stopThr=1e-9,

r"""

Solve the entropic regularization optimal transport problem and return the OT matrix

The function solves the following optimization problem:

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg}\cdot\Omega(\gamma)

s.t. \ \gamma \mathbf{1} &= \mathbf{a}

\gamma^T \mathbf{1} &= \mathbf{b}

\gamma &\geq 0

where :

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

- :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

----------

a : array-like, shape (dim_a,)

samples weights in the source domain

b : array-like, shape (dim_b,) or array-like, shape (dim_b, n_hists)

samples in the target domain, compute sinkhorn with multiple targets

and fixed :math:`\mathbf{M}` if :math:`\mathbf{b}` is a matrix

(return OT loss + dual variables in log)

M : array-like, shape (dim_a, dim_b)

loss matrix

reg : float

Regularization term >0

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 (dim_a, dim_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

-------

gamma : array-like, shape (dim_a, dim_b)

Optimal transportation matrix for the given parameters

log : dict

log dictionary return only if log==True in parameters

Examples

--------

>>> import ot

>>> a=[.5, .5]

>>> b=[.5, .5]

>>> M=[[0., 1.], [1., 0.]]

>>> ot.sinkhorn(a, b, M, 1)

array([[0.36552929, 0.13447071],

[0.13447071, 0.36552929]])

.. _references-sinkhorn-knopp:

References

----------

.. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation

of Optimal Transport, Advances in Neural Information

Processing Systems (NIPS) 26, 2013

See Also

--------

ot.lp.emd : Unregularized OT

ot.optim.cg : General regularized OT

"""

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

if len(a) == 0:

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

if len(b) == 0:

b = nx.full((M.shape[1],), 1.0 / M.shape[1], type_as=M)

# init data

dim_a = len(a)

dim_b = b.shape[0]

if len(b.shape) > 1:

n_hists = b.shape[1]

else:

n_hists = 0

if log:

log = {'err': []}

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

# distances

if warmstart is None:

if n_hists:

u = nx.ones((dim_a, n_hists), type_as=M) / dim_a

v = nx.ones((dim_b, n_hists), type_as=M) / dim_b

else:

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

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

else:

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

K = nx.exp(M / (-reg))

Kp = (1 / a).reshape(-1, 1) * K

err = 1

for ii in range(numItermax):

uprev = u

vprev = v

KtransposeU = nx.dot(K.T, u)

v = b / KtransposeU

u = 1. / nx.dot(Kp, v)

if (nx.any(KtransposeU == 0)

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

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

if n_hists:

tmp2 = nx.einsum('ik,ij,jk->jk', u, K, v)

else:

# compute right marginal tmp2= (diag(u)Kdiag(v))^T1

tmp2 = nx.einsum('i,ij,j->j', u, K, v)

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

if log:

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:

log['niter'] = ii

log['u'] = u

log['v'] = v

if n_hists: # return only loss

res = nx.einsum('ik,ij,jk,ij->k', u, K, v, M)

if log:

return res, log

else:

return res

else: # return OT matrix

if log:

return u.reshape((-1, 1)) * K * v.reshape((1, -1)), log

else:

<div class="viewcode-block" id="sinkhorn_log"><a class="viewcode-back" href="../../gen_modules/ot.bregman.html#ot.sinkhorn_log">[docs]</a>def sinkhorn_log(a, b, M, reg, numItermax=1000, stopThr=1e-9, verbose=False,

log=False, warn=True, warmstart=None, **kwargs):

r"""

Solve the entropic regularization optimal transport problem in log space

and return the OT matrix

The function solves the following optimization problem:

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg}\cdot\Omega(\gamma)

s.t. \ \gamma \mathbf{1} &= \mathbf{a}

\gamma^T \mathbf{1} &= \mathbf{b}

\gamma &\geq 0

where :

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

- :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 :ref:`[2] <references-sinkhorn-log>` with the

implementation from :ref:`[34] <references-sinkhorn-log>`

Parameters

----------

a : array-like, shape (dim_a,)

samples weights in the source domain

b : array-like, shape (dim_b,) or array-like, shape (dim_b, n_hists)

samples in the target domain, compute sinkhorn with multiple targets

and fixed :math:`\mathbf{M}` if :math:`\mathbf{b}` is a matrix (return OT loss + dual variables in log)

M : array-like, shape (dim_a, dim_b)

loss matrix

reg : float

Regularization term >0

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 (dim_a, dim_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

-------

gamma : array-like, shape (dim_a, dim_b)

Optimal transportation matrix for the given parameters

log : dict

log dictionary return only if log==True in parameters

Examples

--------

>>> import ot

>>> a=[.5, .5]

>>> b=[.5, .5]

>>> M=[[0., 1.], [1., 0.]]

>>> ot.sinkhorn(a, b, M, 1)

array([[0.36552929, 0.13447071],

[0.13447071, 0.36552929]])

.. _references-sinkhorn-log:

References

----------

.. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of

Optimal Transport, Advances in Neural Information Processing

Systems (NIPS) 26, 2013

.. [34] Feydy, J., Séjourné, T., Vialard, F. X., Amari, S. I.,

Trouvé, A., & Peyré, G. (2019, April). Interpolating between

optimal transport and MMD using Sinkhorn divergences. In The

22nd International Conference on Artificial Intelligence and

Statistics (pp. 2681-2690). PMLR.

See Also

--------

ot.lp.emd : Unregularized OT

ot.optim.cg : General regularized OT

"""

# init data

dim_a = len(a)

if len(b.shape) > 1:

else:

n_hists = 0

# in case of multiple historgrams

if n_hists > 1 and warmstart is None:

warmstart = [None] * n_hists

if n_hists: # we do not want to use tensors sor we do a loop

lst_loss = []

lst_u = []

lst_v = []

for k in range(n_hists):

res = sinkhorn_log(a, b[:, k], M, reg, numItermax=numItermax, stopThr=stopThr,

verbose=verbose, log=log, warmstart=warmstart[k], **kwargs)

if log:

lst_loss.append(nx.sum(M * res[0]))

lst_u.append(res[1]['log_u'])

lst_v.append(res[1]['log_v'])

else:

res = nx.stack(lst_loss)

if log:

log = {'log_u': nx.stack(lst_u, 1),

'log_v': nx.stack(lst_v, 1), }

log['u'] = nx.exp(log['log_u'])

log['v'] = nx.exp(log['log_v'])

return res, log

else:

return res

else:

if log:

log = {'err': []}

Mr = - M / reg

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

# distances

if warmstart is None:

u = nx.zeros(dim_a, type_as=M)

v = nx.zeros(dim_b, type_as=M)

else:

u, v = warmstart

def get_logT(u, v):

if n_hists:

return Mr[:, :, None] + u + v

else:

return Mr + u[:, None] + v[None, :]

loga = nx.log(a)

logb = nx.log(b)

err = 1

for ii in range(numItermax):

v = logb - nx.logsumexp(Mr + u[:, None], 0)

u = loga - nx.logsumexp(Mr + v[None, :], 1)

if ii % 10 == 0:

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

# the 10th iterations

# compute right marginal tmp2= (diag(u)Kdiag(v))^T1

tmp2 = nx.sum(nx.exp(get_logT(u, v)), 0)

if log:

if verbose:

if ii % 200 == 0:

print(

if err < stopThr:

break

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:

log['niter'] = ii

log['log_u'] = u

log['log_v'] = v

return nx.exp(get_logT(u, v)), log

else:

<div class="viewcode-block" id="greenkhorn"><a class="viewcode-back" href="../../gen_modules/ot.bregman.html#ot.greenkhorn">[docs]</a>def greenkhorn(a, b, M, reg, numItermax=10000, stopThr=1e-9, verbose=False,

r"""

Solve the entropic regularization optimal transport problem and return the OT matrix

The algorithm used is based on the paper :ref:`[22] <references-greenkhorn>`

which is a stochastic version of the Sinkhorn-Knopp

algorithm :ref:`[2] <references-greenkhorn>`

The function solves the following optimization problem:

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg}\cdot\Omega(\gamma)

s.t. \ \gamma \mathbf{1} &= \mathbf{a}

\gamma^T \mathbf{1} &= \mathbf{b}

\gamma &\geq 0

where :

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

- :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)

Parameters

----------

a : array-like, shape (dim_a,)

samples weights in the source domain

b : array-like, shape (dim_b,) or array-like, shape (dim_b, n_hists)

samples in the target domain, compute sinkhorn with multiple targets

and fixed :math:`\mathbf{M}` if :math:`\mathbf{b}` is a matrix

(return OT loss + dual variables in log)

M : array-like, shape (dim_a, dim_b)

loss matrix

reg : float

Regularization term >0

numItermax : int, optional

Max number of iterations

stopThr : float, optional

Stop threshold on error (>0)

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 (dim_a, dim_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

-------

gamma : array-like, shape (dim_a, dim_b)

Optimal transportation matrix for the given parameters

log : dict

log dictionary return only if log==True in parameters

Examples

--------

>>> import ot

>>> a=[.5, .5]

>>> b=[.5, .5]

>>> M=[[0., 1.], [1., 0.]]

>>> ot.bregman.greenkhorn(a, b, M, 1)

array([[0.36552929, 0.13447071],

[0.13447071, 0.36552929]])

.. _references-greenkhorn:

References

----------

.. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation

of Optimal Transport, Advances in Neural Information

Processing Systems (NIPS) 26, 2013

.. [22] J. Altschuler, J.Weed, P. Rigollet : Near-linear time

approximation algorithms for optimal transport via Sinkhorn

iteration, Advances in Neural Information Processing

Systems (NIPS) 31, 2017

See Also

--------

ot.lp.emd : Unregularized OT

ot.optim.cg : General regularized OT

"""

if nx.__name__ in ("jax", "tf"):

raise TypeError("JAX or TF arrays have been received. Greenkhorn is not "

"compatible with neither JAX nor TF")

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

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

dim_a = a.shape[0]

K = nx.exp(-M / reg)

if warmstart is None:

u = nx.full((dim_a,), 1. / dim_a, type_as=K)

v = nx.full((dim_b,), 1. / dim_b, type_as=K)

else:

G = u[:, None] * K * v[None, :]

viol = nx.sum(G, axis=1) - a

viol_2 = nx.sum(G, axis=0) - b

stopThr_val = 1

if log:

log = dict()

log['u'] = u

log['v'] = v

for ii in range(numItermax):

i_1 = nx.argmax(nx.abs(viol))

i_2 = nx.argmax(nx.abs(viol_2))

m_viol_1 = nx.abs(viol[i_1])

m_viol_2 = nx.abs(viol_2[i_2])

stopThr_val = nx.maximum(m_viol_1, m_viol_2)

if m_viol_1 > m_viol_2:

old_u = u[i_1]

new_u = a[i_1] / nx.dot(K[i_1, :], v)

G[i_1, :] = new_u * K[i_1, :] * v

viol[i_1] = nx.dot(new_u * K[i_1, :], v) - a[i_1]

viol_2 += (K[i_1, :].T * (new_u - old_u) * v)

u[i_1] = new_u

else:

old_v = v[i_2]

new_v = b[i_2] / nx.dot(K[:, i_2].T, u)

G[:, i_2] = u * K[:, i_2] * new_v

# aviol = (G@one_m - a)

# aviol_2 = (G.T@one_n - b)

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