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<!DOCTYPE html>

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

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

"""

Regularized Unbalanced OT solvers

"""

# Author: Hicham Janati <hicham.janati@inria.fr>

# Laetitia Chapel <laetitia.chapel@univ-ubs.fr>

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

#

# License: MIT License

from __future__ import division

import warnings

import numpy as np

from scipy.optimize import minimize, Bounds

from .backend import get_backend

from .utils import list_to_array, get_parameter_pair

def sinkhorn_unbalanced(a, b, M, reg, reg_m, method='sinkhorn',

reg_type="entropy", warmstart=None, numItermax=1000,

stopThr=1e-6, verbose=False, log=False, **kwargs):

r"""

Solve the unbalanced entropic regularization optimal transport problem

and return the OT plan

The function solves the following optimization problem:

.. math::

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

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

\mathrm{reg_{m1}} \cdot \mathrm{KL}(\gamma \mathbf{1}, \mathbf{a}) +

\mathrm{reg_{m2}} \cdot \mathrm{KL}(\gamma^T \mathbf{1}, \mathbf{b})

s.t.

\gamma \geq 0

where :

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

- :math:`\Omega` is the entropic regularization term, can be either KL divergence or negative entropy

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

- KL is the Kullback-Leibler divergence

The algorithm used for solving the problem is the generalized

Sinkhorn-Knopp matrix scaling algorithm as proposed in :ref:`[10, 25] <references-sinkhorn-unbalanced>`

Parameters

----------

a : array-like (dim_a,)

Unnormalized histogram of dimension `dim_a`

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

One or multiple unnormalized histograms of dimension `dim_b`.

If many, compute all the OT distances :math:`(\mathbf{a}, \mathbf{b}_i)_i`

M : array-like (dim_a, dim_b)

loss matrix

reg : float

Entropy regularization term > 0

reg_m: float or indexable object of length 1 or 2

Marginal relaxation term.

If reg_m is a scalar or an indexable object of length 1,

then the same reg_m is applied to both marginal relaxations.

The entropic balanced OT can be recovered using `reg_m=float("inf")`.

For semi-relaxed case, use either

`reg_m=(float("inf"), scalar)` or `reg_m=(scalar, float("inf"))`.

If reg_m is an array, it must have the same backend as input arrays (a, b, M).

method : str

method used for the solver either 'sinkhorn', 'sinkhorn_stabilized' or

'sinkhorn_reg_scaling', see those function for specific parameters

reg_type : string, optional

Regularizer term. Can take two values:

'entropy' (negative entropy)

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

'kl' (Kullback-Leibler)

:math:`\Omega(\gamma) = \text{KL}(\gamma, \mathbf{a} \mathbf{b}^T)`.

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).

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

Returns

-------

if n_hists == 1:

- gamma : (dim_a, dim_b) array-like

Optimal transportation matrix for the given parameters

- log : dict

log dictionary returned only if `log` is `True`

else:

- ot_distance : (n_hists,) array-like

the OT distance between :math:`\mathbf{a}` and each of the histograms :math:`\mathbf{b}_i`

- log : dict

log dictionary returned only if `log` is `True`

Examples

--------

>>> import ot

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

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

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

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

array([[0.51122814, 0.18807032],

[0.18807032, 0.51122814]])

.. _references-sinkhorn-unbalanced:

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.

.. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. :

Learning with a Wasserstein Loss, Advances in Neural Information

Processing Systems (NIPS) 2015

See Also

--------

ot.unbalanced.sinkhorn_knopp_unbalanced : Unbalanced Classic Sinkhorn :ref:`[10] <references-sinkhorn-unbalanced>`

ot.unbalanced.sinkhorn_stabilized_unbalanced:

Unbalanced Stabilized sinkhorn :ref:`[9, 10] <references-sinkhorn-unbalanced>`

ot.unbalanced.sinkhorn_reg_scaling_unbalanced:

Unbalanced Sinkhorn with epsilon scaling :ref:`[9, 10] <references-sinkhorn-unbalanced>`

"""

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

return sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, reg_type,

warmstart, numItermax=numItermax,

stopThr=stopThr, verbose=verbose,

log=log, **kwargs)

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

return sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, reg_type,

warmstart, numItermax=numItermax,

stopThr=stopThr,

verbose=verbose,

log=log, **kwargs)

elif method.lower() in ['sinkhorn_reg_scaling']:

warnings.warn('Method not implemented yet. Using classic Sinkhorn-Knopp')

warmstart, numItermax=numItermax,

log=log, **kwargs)

else:

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

def sinkhorn_unbalanced2(a, b, M, reg, reg_m, method='sinkhorn',

r"""

Solve the entropic regularization unbalanced 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) +

\mathrm{reg_{m1}} \cdot \mathrm{KL}(\gamma \mathbf{1}, \mathbf{a}) +

\mathrm{reg_{m2}} \cdot \mathrm{KL}(\gamma^T \mathbf{1}, \mathbf{b})

s.t.

\gamma\geq 0

where :

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

- :math:`\Omega` is the entropic regularization term, can be either KL divergence or negative entropy

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

- KL is the Kullback-Leibler divergence

The algorithm used for solving the problem is the generalized

Sinkhorn-Knopp matrix scaling algorithm as proposed in :ref:`[10, 25] <references-sinkhorn-unbalanced2>`

Parameters

----------

a : array-like (dim_a,)

Unnormalized histogram of dimension `dim_a`

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

One or multiple unnormalized histograms of dimension `dim_b`.

If many, compute all the OT distances :math:`(\mathbf{a}, \mathbf{b}_i)_i`

M : array-like (dim_a, dim_b)

loss matrix

reg : float

Entropy regularization term > 0

reg_m: float or indexable object of length 1 or 2

Marginal relaxation term.

If reg_m is a scalar or an indexable object of length 1,

then the same reg_m is applied to both marginal relaxations.

The entropic balanced OT can be recovered using `reg_m=float("inf")`.

For semi-relaxed case, use either

`reg_m=(float("inf"), scalar)` or `reg_m=(scalar, float("inf"))`.

If reg_m is an array, it must have the same backend as input arrays (a, b, M).

method : str

method used for the solver either 'sinkhorn', 'sinkhorn_stabilized' or

'sinkhorn_reg_scaling', see those function for specific parameterss

reg_type : string, optional

Regularizer term. Can take two values:

'entropy' (negative entropy)

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

'kl' (Kullback-Leibler)

:math:`\Omega(\gamma) = \text{KL}(\gamma, \mathbf{a} \mathbf{b}^T)`.

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).

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

Returns

-------

ot_distance : (n_hists,) array-like

the OT distance between :math:`\mathbf{a}` and each of the histograms :math:`\mathbf{b}_i`

log : dict

log dictionary returned only if `log` is `True`

Examples

--------

>>> import ot

>>> import numpy as np

>>> a=[.5, .10]

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

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

>>> np.round(ot.unbalanced.sinkhorn_unbalanced2(a, b, M, 1., 1.), 8)

0.31912858

.. _references-sinkhorn-unbalanced2:

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.

.. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. :

Learning with a Wasserstein Loss, Advances in Neural Information

Processing Systems (NIPS) 2015

See Also

--------

ot.unbalanced.sinkhorn_knopp : Unbalanced Classic Sinkhorn :ref:`[10] <references-sinkhorn-unbalanced2>`

ot.unbalanced.sinkhorn_stabilized: Unbalanced Stabilized sinkhorn :ref:`[9, 10] <references-sinkhorn-unbalanced2>`

ot.unbalanced.sinkhorn_reg_scaling: Unbalanced Sinkhorn with epsilon scaling :ref:`[9, 10] <references-sinkhorn-unbalanced2>`

"""

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

nx = get_backend(M, a, b)

if len(b.shape) < 2:

res = sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, reg_type,

warmstart, numItermax=numItermax,

log=log, **kwargs)

res = sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, reg_type,

warmstart, numItermax=numItermax,

log=log, **kwargs)

warmstart, numItermax=numItermax,

log=log, **kwargs)

else:

raise ValueError('Unknown method %s.' % method)

if log:

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

else:

warmstart, numItermax=numItermax,

log=log, **kwargs)

warmstart, numItermax=numItermax,

log=log, **kwargs)

warmstart, numItermax=numItermax,

log=log, **kwargs)

else:

def sinkhorn_knopp_unbalanced(a, b, M, reg, reg_m, reg_type="entropy",

warmstart=None, numItermax=1000, stopThr=1e-6,

verbose=False, log=False, **kwargs):

r"""

Solve the entropic regularization unbalanced optimal transport problem and

return the OT plan

The function solves the following optimization problem:

.. math::

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

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

\mathrm{reg_{m1}} \cdot \mathrm{KL}(\gamma \mathbf{1}, \mathbf{a}) +

\mathrm{reg_{m2}} \cdot \mathrm{KL}(\gamma^T \mathbf{1}, \mathbf{b})

s.t.

\gamma \geq 0

where :

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

- :math:`\Omega` is the entropic regularization term, can be either KL divergence or negative entropy

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

- KL is the Kullback-Leibler divergence

The algorithm used for solving the problem is the generalized Sinkhorn-Knopp matrix scaling algorithm as proposed in :ref:`[10, 25] <references-sinkhorn-knopp-unbalanced>`

Parameters

----------

a : array-like (dim_a,)

Unnormalized histogram of dimension `dim_a`

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

One or multiple unnormalized histograms of dimension `dim_b`

If many, compute all the OT distances (a, b_i)

M : array-like (dim_a, dim_b)

loss matrix

reg : float

Entropy regularization term > 0

reg_m: float or indexable object of length 1 or 2

Marginal relaxation term.

If reg_m is a scalar or an indexable object of length 1,

then the same reg_m is applied to both marginal relaxations.

The entropic balanced OT can be recovered using `reg_m=float("inf")`.

For semi-relaxed case, use either

`reg_m=(float("inf"), scalar)` or `reg_m=(scalar, float("inf"))`.

If reg_m is an array, it must have the same backend as input arrays (a, b, M).

reg_type : string, optional

Regularizer term. Can take two values:

'entropy' (negative entropy)

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

'kl' (Kullback-Leibler)

:math:`\Omega(\gamma) = \text{KL}(\gamma, \mathbf{a} \mathbf{b}^T)`.

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).

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

Returns

-------

if n_hists == 1:

- gamma : (dim_a, dim_b) array-like

Optimal transportation matrix for the given parameters

- log : dict

log dictionary returned only if `log` is `True`

else:

- ot_distance : (n_hists,) array-like

the OT distance between :math:`\mathbf{a}` and each of the histograms :math:`\mathbf{b}_i`

- log : dict

log dictionary returned only if `log` is `True`

Examples

--------

>>> import ot

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

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

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

>>> ot.unbalanced.sinkhorn_knopp_unbalanced(a, b, M, 1., 1.)

array([[0.51122814, 0.18807032],

[0.18807032, 0.51122814]])

.. _references-sinkhorn-knopp-unbalanced:

References

----------

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

Scaling algorithms for unbalanced transport problems. arXiv preprint

arXiv:1607.05816.

.. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. :

Learning with a Wasserstein Loss, Advances in Neural Information

Processing Systems (NIPS) 2015

See Also

--------

ot.lp.emd : Unregularized OT

ot.optim.cg : General regularized OT

"""

dim_a, dim_b = M.shape

if len(a) == 0:

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

if len(b) == 0:

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

if len(b.shape) > 1:

n_hists = b.shape[1]

else:

n_hists = 0

reg_m1, reg_m2 = get_parameter_pair(reg_m)

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, 1), type_as=M)

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

a = a.reshape(dim_a, 1)

else:

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

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

else:

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

if reg_type == "kl":

K = nx.exp(-M / reg) * a.reshape(-1)[:, None] * b.reshape(-1)[None, :]

elif reg_type == "entropy":

fi_1 = reg_m1 / (reg_m1 + reg) if reg_m1 != float("inf") else 1

fi_2 = reg_m2 / (reg_m2 + reg) if reg_m2 != float("inf") else 1

err = 1.

for i in range(numItermax):

uprev = u

vprev = v

Kv = nx.dot(K, v)

u = (a / Kv) ** fi_1

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

v = (b / Ktu) ** fi_2

if (nx.any(Ktu == 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('Numerical errors at iteration %s' % i)

u = uprev

v = vprev

break

err_u = nx.max(nx.abs(u - uprev)) / max(

nx.max(nx.abs(u)), nx.max(nx.abs(uprev)), 1.

)

err_v = nx.max(nx.abs(v - vprev)) / max(

nx.max(nx.abs(v)), nx.max(nx.abs(vprev)), 1.

)

err = 0.5 * (err_u + err_v)

if log:

log['err'].append(err)

if verbose:

if i % 50 == 0:

print(

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

print('{:5d}|{:8e}|'.format(i, err))

if err < stopThr:

break

if log:

log['logu'] = nx.log(u + 1e-300)

log['logv'] = nx.log(v + 1e-300)

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[:, None] * K * v[None, :], log

else:

def sinkhorn_stabilized_unbalanced(a, b, M, reg, reg_m, reg_type="entropy",

warmstart=None, tau=1e5,

numItermax=1000, stopThr=1e-6,

r"""

Solve the entropic regularization unbalanced optimal transport

problem and return the loss

The function solves the following optimization problem using log-domain

stabilization as proposed in :ref:`[10] <references-sinkhorn-stabilized-unbalanced>`:

.. math::

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

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

\mathrm{reg_{m1}} \cdot \mathrm{KL}(\gamma \mathbf{1}, \mathbf{a}) +

\mathrm{reg_{m2}} \cdot \mathrm{KL}(\gamma^T \mathbf{1}, \mathbf{b})

s.t.

\gamma \geq 0

where :

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

- :math:`\Omega` is the entropic regularization term, can be either KL divergence or negative entropy

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

- KL is the Kullback-Leibler divergence

The algorithm used for solving the problem is the generalized

Sinkhorn-Knopp matrix scaling algorithm as proposed in :ref:`[10, 25] <references-sinkhorn-stabilized-unbalanced>`

Parameters

----------

a : array-like (dim_a,)

Unnormalized histogram of dimension `dim_a`

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

One or multiple unnormalized histograms of dimension `dim_b`.

If many, compute all the OT distances :math:`(\mathbf{a}, \mathbf{b}_i)_i`

M : array-like (dim_a, dim_b)

loss matrix

reg : float

Entropy regularization term > 0

reg_m: float or indexable object of length 1 or 2

Marginal relaxation term.

If reg_m is a scalar or an indexable object of length 1,

then the same reg_m is applied to both marginal relaxations.

The entropic balanced OT can be recovered using `reg_m=float("inf")`.

For semi-relaxed case, use either

`reg_m=(float("inf"), scalar)` or `reg_m=(scalar, float("inf"))`.

If reg_m is an array, it must have the same backend as input arrays (a, b, M).

reg_type : string, optional

Regularizer term. Can take two values:

'entropy' (negative entropy)

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

'kl' (Kullback-Leibler)

:math:`\Omega(\gamma) = \text{KL}(\gamma, \mathbf{a} \mathbf{b}^T)`.

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).

tau : float

threshold for max value in u or v for log scaling

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

Returns

-------

if n_hists == 1:

- gamma : (dim_a, dim_b) array-like

Optimal transportation matrix for the given parameters

- log : dict

log dictionary returned only if `log` is `True`

else:

- ot_distance : (n_hists,) array-like

the OT distance between :math:`\mathbf{a}` and each of the histograms :math:`\mathbf{b}_i`

- log : dict

log dictionary returned only if `log` is `True`

Examples

--------

>>> import ot

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

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

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

>>> ot.unbalanced.sinkhorn_stabilized_unbalanced(a, b, M, 1., 1.)

array([[0.51122814, 0.18807032],

[0.18807032, 0.51122814]])

.. _references-sinkhorn-stabilized-unbalanced:

References

----------

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

Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.

.. [25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. :

Learning with a Wasserstein Loss, Advances in Neural Information

Processing Systems (NIPS) 2015

See Also

--------

ot.lp.emd : Unregularized OT

ot.optim.cg : General regularized OT

"""

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

dim_a, dim_b = M.shape

if len(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)

else:

if reg_type == "kl":

log_ab = nx.log(a + 1e-16).reshape(-1)[:, None] + nx.log(b + 1e-16).reshape(-1)[None, :]

M0 = M - reg * log_ab

else:

M0 = M

cpt = 0

err = 1.

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

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

ones_a = nx.ones(dim_a, type_as=M)

ones_b = nx.ones(dim_b, type_as=M)

while (err > stopThr and cpt < numItermax):

uprev = u

vprev = v

f_alpha = nx.exp(- alpha / (reg + reg_m1)) if reg_m1 != float("inf") else ones_a

f_beta = nx.exp(- beta / (reg + reg_m2)) if reg_m2 != float("inf") else ones_b

if n_hists:

f_alpha = f_alpha[:, None]

f_beta = f_beta[:, None]

u = ((a / (Kv + 1e-16)) ** fi_1) * f_alpha

v = ((b / (Ktu + 1e-16)) ** fi_2) * f_beta

absorbing = False

if nx.any(u > tau) or nx.any(v > tau):

absorbing = True

if n_hists:

alpha = alpha + reg * nx.log(nx.max(u, 1))

beta = beta + reg * nx.log(nx.max(v, 1))

else:

K = nx.exp((alpha[:, None] + beta[None, :] - M0) / reg)

v = nx.ones(v.shape, type_as=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

u = uprev

v = vprev

break

if (cpt % 10 == 0 and not absorbing) or cpt == 0:

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

# the 10th iterations

err = nx.max(nx.abs(u - uprev)) / max(

)

if log:

if verbose:

if cpt % 200 == 0:

print(

print('{:5d}|{:8e}|'.format(cpt, err))

cpt = cpt + 1

if err > stopThr:

warnings.warn("Stabilized Unbalanced Sinkhorn did not converge." +

"Try a larger entropy `reg` or a lower mass `reg_m`." +

"Or a larger absorption threshold `tau`.")

if n_hists:

logu = alpha[:, None] / reg + nx.log(u)

logv = beta[:, None] / reg + nx.log(v)

else:

logu = alpha / reg + nx.log(u)

logv = beta / reg + nx.log(v)

if log:

log['logu'] = logu

log['logv'] = logv

if n_hists: # return only loss

res = nx.logsumexp(

nx.log(M + 1e-100)[:, :, None]

+ logu[:, None, :]

+ logv[None, :, :]

- M0[:, :, None] / reg,

axis=(0, 1)

)

res = nx.exp(res)

if log:

return res, log

else:

return res

else: # return OT matrix

ot_matrix = nx.exp(logu[:, None] + logv[None, :] - M0 / reg)

if log:

return ot_matrix, log

else:

return ot_matrix</div>

def barycenter_unbalanced_stabilized(A, M, reg, reg_m, weights=None, tau=1e3,

r"""Compute the entropic unbalanced wasserstein barycenter of :math:`\mathbf{A}` with stabilization.

The function solves the following optimization problem:

.. math::

\mathbf{a} = \mathop{\arg \min}_\mathbf{a} \quad \sum_i W_{u_{reg}}(\mathbf{a},\mathbf{a}_i)

where :

- :math:`W_{u_{reg}}(\cdot,\cdot)` is the unbalanced entropic regularized Wasserstein distance (see :py:func:`ot.unbalanced.sinkhorn_unbalanced`)

- :math:`\mathbf{a}_i` are training distributions in the columns of matrix :math:`\mathbf{A}`

- reg and :math:`\mathbf{M}` are respectively the regularization term and the cost matrix for OT

- reg_mis the marginal relaxation hyperparameter

The algorithm used for solving the problem is the generalized

Sinkhorn-Knopp matrix scaling algorithm as proposed in :ref:`[10] <references-barycenter-unbalanced-stabilized>`

Parameters

----------

A : array-like (dim, n_hists)

`n_hists` training distributions :math:`\mathbf{a}_i` of dimension `dim`

M : array-like (dim, dim)

ground metric matrix for OT.

reg : float

Entropy regularization term > 0

reg_m : float

Marginal relaxation term > 0

tau : float

Stabilization threshold for log domain absorption.

weights : array-like (n_hists,) optional

Weight of each distribution (barycentric coordinates)

If None, uniform weights are used.

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

Returns

-------

a : (dim,) array-like

Unbalanced Wasserstein barycenter

log : dict

log dictionary return only if log==True in parameters

.. _references-barycenter-unbalanced-stabilized:

References

----------

.. [3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré,

G. (2015). Iterative Bregman projections for regularized transportation

problems. SIAM Journal on Scientific Computing, 37(2), A1111-A1138.

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

Scaling algorithms for unbalanced transport problems. arXiv preprint

arXiv:1607.05816.

"""

A, M = list_to_array(A, M)

nx = get_backend(A, M)

dim, n_hists = A.shape

if weights is None:

weights = nx.ones(n_hists, type_as=A) / n_hists

else:

assert (len(weights) == A.shape[1])

if log:

log = {'err': []}

fi = reg_m / (reg_m + reg)

u = nx.ones((dim, n_hists), type_as=A) / dim

v = nx.ones((dim, n_hists), type_as=A) / dim

# print(reg)

cpt = 0

err = 1.

alpha = nx.zeros(dim, type_as=A)

beta = nx.zeros(dim, type_as=A)

q = nx.ones(dim, type_as=A) / dim

for i in range(numItermax):

qprev = nx.copy(q)

f_alpha = f_alpha[:, None]

f_beta = f_beta[:, None]

u = ((A / (Kv + 1e-16)) ** fi) * f_alpha

q = (Ktu ** (1 - fi)) * f_beta

q = nx.dot(q, weights) ** (1 / (1 - fi))

Q = q[:, None]

v = ((Q / (Ktu + 1e-16)) ** fi) * f_beta

absorbing = False

absorbing = True

K = nx.exp((alpha[:, None] + beta[None, :] - M) / reg)

v = nx.ones(v.shape, type_as=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

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