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<!DOCTYPE html>

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

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<h1>Source code for ot.unbalanced._mm</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 ..backend import get_backend

from ..utils import list_to_array, get_parameter_pair

def mm_unbalanced(

a,

b,

M,

reg_m,

c=None,

reg=0,

div="kl",

G0=None,

numItermax=1000,

stopThr=1e-15,

verbose=False,

log=False,

):

r"""

Solve the unbalanced optimal transport problem and return the OT plan.

The function solves the following optimization problem:

.. math::

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

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

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

\mathrm{reg} \cdot \mathrm{div}(\gamma, \mathbf{c})

s.t.

\gamma \geq 0

where:

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

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

unbalanced distributions

- :math:`\mathbf{c}` is a reference distribution for the regularization

- div is a divergence, either Kullback-Leibler or half-squared :math:`\ell_2` divergence

The algorithm used for solving the problem is a maximization-

minimization algorithm as proposed in :ref:`[41] <references-regpath>`

Parameters

----------

a : array-like (dim_a,)

Unnormalized histogram of dimension `dim_a`

If `a` is an empty list or array ([]),

then `a` is set to uniform distribution.

b : array-like (dim_b,)

Unnormalized histogram of dimension `dim_b`

If `b` is an empty list or array ([]),

then `b` is set to uniform distribution.

M : array-like (dim_a, dim_b)

loss matrix

reg_m: float or indexable object of length 1 or 2

Marginal relaxation term: nonnegative but cannot be infinity.

If :math:`\mathrm{reg_{m}}` is a scalar or an indexable object of length 1,

then the same :math:`\mathrm{reg_{m}}` is applied to both marginal relaxations.

If :math:`\mathrm{reg_{m}}` is an array,

it must have the same backend as input arrays `(a, b, M)`.

reg : float, optional (default = 0)

Regularization term >= 0.

By default, solve the unregularized problem

c : array-like (dim_a, dim_b), optional (default = None)

Reference measure for the regularization.

If None, then use :math:`\mathbf{c} = \mathbf{a} \mathbf{b}^T`.

div: string, optional

Divergence to quantify the difference between the marginals.

Can take two values: 'kl' (Kullback-Leibler) or 'l2' (half-squared)

G0: array-like (dim_a, dim_b)

Initialization of the transport matrix

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

-------

gamma : (dim_a, dim_b) array-like

Optimal transportation matrix for the given parameters

log : dict

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

Examples

--------

>>> import ot

>>> import numpy as np

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

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

>>> M=[[1., 36.],[9., 4.]]

>>> np.round(ot.unbalanced.mm_unbalanced(a, b, M, 5, div='kl'), 2)

array([[0.45, 0. ],

[0. , 0.34]])

>>> np.round(ot.unbalanced.mm_unbalanced(a, b, M, 5, div='l2'), 2)

array([[0.4, 0. ],

[0. , 0.1]])

.. _references-regpath:

References

----------

.. [41] Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

See Also

--------

ot.lp.emd : Unregularized OT

ot.unbalanced.sinkhorn_unbalanced : Entropic regularized OT

"""

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

nx = get_backend(M, a, b)

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

G = a[:, None] * b[None, :] if G0 is None else G0

if reg > 0: # regularized case

c = a[:, None] * b[None, :] if c is None else c

else: # unregularized case

c = 0

reg_m1, reg_m2 = get_parameter_pair(reg_m)

if log:

log = {"err": [], "G": []}

div = div.lower()

if div == "kl":

sum_r = reg + reg_m1 + reg_m2

r1, r2, r = reg_m1 / sum_r, reg_m2 / sum_r, reg / sum_r

K = (a[:, None] ** r1) * (b[None, :] ** r2) * (c**r) * nx.exp(-M / sum_r)

elif div == "l2":

K = (reg_m1 * a[:, None]) + (reg_m2 * b[None, :]) + reg * c - M

K = nx.maximum(K, nx.zeros((dim_a, dim_b), type_as=M))

else:

raise ValueError("Unknown div = {}. Must be either 'kl' or 'l2'".format(div))

for i in range(numItermax):

Gprev = G

if div == "kl":

Gd = (nx.sum(G, 1, keepdims=True) ** r1) * (

nx.sum(G, 0, keepdims=True) ** r2

) + 1e-16

G = K * G ** (r1 + r2) / Gd

elif div == "l2":

Gd = (

reg_m1 * nx.sum(G, 1, keepdims=True)

+ reg_m2 * nx.sum(G, 0, keepdims=True)

+ reg * G

+ 1e-16

)

G = K * G / Gd

err = nx.sqrt(nx.sum((G - Gprev) ** 2))

if log:

log["err"].append(err)

log["G"].append(G)

if verbose:

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

if err < stopThr:

break

if log:

linear_cost = nx.sum(G * M)

log["cost"] = linear_cost

m1, m2 = nx.sum(G, 1), nx.sum(G, 0)

if div == "kl":

cost = (

linear_cost

+ reg_m1 * nx.kl_div(m1, a, mass=True)

+ reg_m2 * nx.kl_div(m2, b, mass=True)

)

if reg > 0:

cost = cost + reg * nx.kl_div(G, c, mass=True)

else:

cost = (

linear_cost

+ reg_m1 * 0.5 * nx.sum((m1 - a) ** 2)

+ reg_m2 * 0.5 * nx.sum((m2 - b) ** 2)

)

if reg > 0:

cost = cost + reg * 0.5 * nx.sum((G - c) ** 2)

log["total_cost"] = cost

return G, log

else:

return G</div>

def mm_unbalanced2(

a,

b,

M,

reg_m,

c=None,

reg=0,

div="kl",

G0=None,

returnCost="linear",

numItermax=1000,

stopThr=1e-15,

verbose=False,

log=False,

):

r"""

Solve the unbalanced optimal transport problem and return the OT cost.

The function solves the following optimization problem:

.. math::

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

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

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

\mathrm{reg} \cdot \mathrm{div}(\gamma, \mathbf{c})

s.t.

\gamma \geq 0

where:

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

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

unbalanced distributions

- :math:`\mathbf{c}` is a reference distribution for the regularization

- :math:`\mathrm{div}` is a divergence, either Kullback-Leibler or half-squared :math:`\ell_2` divergence

The algorithm used for solving the problem is a maximization-

minimization algorithm as proposed in :ref:`[41] <references-regpath>`

Parameters

----------

a : array-like (dim_a,)

Unnormalized histogram of dimension `dim_a`

If `a` is an empty list or array ([]),

then `a` is set to uniform distribution.

b : array-like (dim_b,)

Unnormalized histogram of dimension `dim_b`

If `b` is an empty list or array ([]),

then `b` is set to uniform distribution.

M : array-like (dim_a, dim_b)

loss matrix

reg_m: float or indexable object of length 1 or 2

Marginal relaxation term: nonnegative but cannot be infinity.

If :math:`\mathrm{reg_{m}}` is a scalar or an indexable object of length 1,

then the same :math:`\mathrm{reg_{m}}` is applied to both marginal relaxations.

If :math:`\mathrm{reg_{m}}` is an array,

it must have the same backend as input arrays `(a, b, M)`.

reg : float, optional (default = 0)

Entropy regularization term >= 0.

By default, solve the unregularized problem

c : array-like (dim_a, dim_b), optional (default = None)

Reference measure for the regularization.

If None, then use :math:`\mathbf{c} = \mathbf{a} \mathbf{b}^T`.

div: string, optional

Divergence to quantify the difference between the marginals.

Can take two values: 'kl' (Kullback-Leibler) or 'l2' (half-squared)

G0: array-like (dim_a, dim_b)

Initialization of the transport matrix

returnCost: string, optional (default = "linear")

If `returnCost` = "linear", then return the linear part of the unbalanced OT loss.

If `returnCost` = "total", then return the total unbalanced OT loss.

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_cost : array-like

the OT cost between :math:`\mathbf{a}` and :math:`\mathbf{b}`

log : dict

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

Examples

--------

>>> import ot

>>> import numpy as np

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

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

>>> M=[[1., 36.],[9., 4.]]

>>> np.round(ot.unbalanced.mm_unbalanced2(a, b, M, 5, div='l2'), 2)

0.8

>>> np.round(ot.unbalanced.mm_unbalanced2(a, b, M, 5, div='kl'), 2)

1.79

References

----------

.. [41] Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

See Also

--------

ot.lp.emd2 : Unregularized OT loss

ot.unbalanced.sinkhorn_unbalanced2 : Entropic regularized OT loss

"""

_, log_mm = mm_unbalanced(

a,

b,

M,

reg_m,

c=c,

reg=reg,

div=div,

G0=G0,

numItermax=numItermax,

stopThr=stopThr,

verbose=verbose,

log=True,

)

if returnCost == "linear":

cost = log_mm["cost"]

elif returnCost == "total":

cost = log_mm["total_cost"]

else:

raise ValueError("Unknown returnCost = {}".format(returnCost))

if log:

return cost, log_mm

else:

return cost</div>

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