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<!DOCTYPE html>

<head>

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

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

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

"""

Partial OT solvers

"""

# Author: Laetitia Chapel <laetitia.chapel@irisa.fr>

# Yikun Bai < yikun.bai@vanderbilt.edu >

# Cédric Vincent-Cuaz <cedvincentcuaz@gmail.com>

from ..utils import list_to_array

from ..backend import get_backend

from ..lp import emd

from .partial_cython import partial_wasserstein_1d_cy

import numpy as np

import warnings

# License: MIT License

def partial_wasserstein_lagrange(

a, b, M, reg_m=None, nb_dummies=1, log=False, **kwargs

):

r"""

Solves the partial optimal transport problem for the quadratic cost

and returns the OT plan

The function considers the following problem:

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, (\mathbf{M} - \lambda) \rangle_F

.. math::

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

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

\gamma &\geq 0

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

\leq \min\{\|\mathbf{a}\|_1, \|\mathbf{b}\|_1\}

or equivalently (see Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X.

(2018). An interpolating distance between optimal transport and Fisher–Rao

metrics. Foundations of Computational Mathematics, 18(1), 1-44.)

.. math::

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

\sqrt{\frac{\lambda}{2} (\|\gamma \mathbf{1} - \mathbf{a}\|_1 + \|\gamma^T \mathbf{1} - \mathbf{b}\|_1)}

s.t. \ \gamma \geq 0

where :

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

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

- :math:`\lambda` is the lagrangian cost. Tuning its value allows attaining

a given mass to be transported `m`

The formulation of the problem has been proposed in

:ref:`[28] <references-partial-wasserstein-lagrange>`

Parameters

----------

a : np.ndarray (dim_a,)

Unnormalized histogram of dimension `dim_a`

b : np.ndarray (dim_b,)

Unnormalized histograms of dimension `dim_b`

M : np.ndarray (dim_a, dim_b)

cost matrix for the quadratic cost

reg_m : float, optional

Lagrangian cost

nb_dummies : int, optional, default:1

number of reservoir points to be added (to avoid numerical

instabilities, increase its value if an error is raised)

log : bool, optional

record log if True

**kwargs : dict

parameters can be directly passed to the emd solver

.. warning::

When dealing with a large number of points, the EMD solver may face

some instabilities, especially when the mass associated to the dummy

point is large. To avoid them, increase the number of dummy points

(allows a smoother repartition of the mass over the points).

Returns

-------

gamma : (dim_a, dim_b) ndarray

Optimal transportation matrix for the given parameters

log : dict

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

Examples

--------

>>> import ot

>>> a = [.1, .2]

>>> b = [.1, .1]

>>> M = [[0., 1.], [2., 3.]]

>>> np.round(partial_wasserstein_lagrange(a,b,M), 2)

array([[0.1, 0. ],

[0. , 0.1]])

>>> np.round(partial_wasserstein_lagrange(a,b,M,reg_m=2), 2)

array([[0.1, 0. ],

[0. , 0. ]])

.. _references-partial-wasserstein-lagrange:

References

----------

.. [28] Caffarelli, L. A., & McCann, R. J. (2010) Free boundaries in

optimal transport and Monge-Ampere obstacle problems. Annals of

mathematics, 673-730.

See Also

--------

ot.partial.partial_wasserstein : Partial Wasserstein with fixed mass

"""

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

nx = get_backend(a, b, M)

if nx.sum(a) > 1 + 1e-15 or nx.sum(b) > 1 + 1e-15: # 1e-15 for numerical errors

raise ValueError("Problem infeasible. Check that a and b are in the simplex")

if reg_m is None:

reg_m = float(nx.max(M)) + 1

if reg_m < -nx.max(M):

return nx.zeros((len(a), len(b)), type_as=M)

a0, b0, M0 = a, b, M

# convert to humpy

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

eps = 1e-20

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

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

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

M_star = M - reg_m # modified cost matrix

# trick to fasten the computation: select only the subset of columns/lines

# that can have marginals greater than 0 (that is to say M < 0)

idx_x = np.where(np.min(M_star, axis=1) < eps)[0]

idx_y = np.where(np.min(M_star, axis=0) < eps)[0]

# extend a, b, M with "reservoir" or "dummy" points

M_extended = np.zeros((len(idx_x) + nb_dummies, len(idx_y) + nb_dummies))

M_extended[: len(idx_x), : len(idx_y)] = M_star[np.ix_(idx_x, idx_y)]

a_extended = np.append(

a[idx_x], [(np.sum(a) - np.sum(a[idx_x]) + np.sum(b)) / nb_dummies] * nb_dummies

)

b_extended = np.append(

b[idx_y], [(np.sum(b) - np.sum(b[idx_y]) + np.sum(a)) / nb_dummies] * nb_dummies

)

gamma_extended, log_emd = emd(

a_extended, b_extended, M_extended, log=True, **kwargs

)

gamma = np.zeros((len(a), len(b)))

gamma[np.ix_(idx_x, idx_y)] = gamma_extended[:-nb_dummies, :-nb_dummies]

# convert back to backend

gamma = nx.from_numpy(gamma, type_as=M0)

if log_emd["warning"] is not None:

raise ValueError(

"Error in the EMD resolution: try to increase the number of dummy points"

)

log_emd["cost"] = nx.sum(gamma * M0)

log_emd["u"] = nx.from_numpy(log_emd["u"], type_as=a0)

log_emd["v"] = nx.from_numpy(log_emd["v"], type_as=b0)

if log:

return gamma, log_emd

else:

return gamma</div>

def partial_wasserstein(a, b, M, m=None, nb_dummies=1, log=False, **kwargs):

r"""

Solves the partial optimal transport problem for the quadratic cost

and returns the OT plan

The function considers the following problem:

.. math::

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

.. math::

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

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

\gamma &\geq 0

\mathbf{1}^T \gamma^T \mathbf{1} = m &\leq \min\{\|\mathbf{a}\|_1, \|\mathbf{b}\|_1\}

where :

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

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

- `m` is the amount of mass to be transported

Parameters

----------

a : np.ndarray (dim_a,)

Unnormalized histogram of dimension `dim_a`

b : np.ndarray (dim_b,)

Unnormalized histograms of dimension `dim_b`

M : np.ndarray (dim_a, dim_b)

cost matrix for the quadratic cost

m : float, optional

amount of mass to be transported

nb_dummies : int, optional, default:1

number of reservoir points to be added (to avoid numerical

instabilities, increase its value if an error is raised)

log : bool, optional

record log if True

**kwargs : dict

parameters can be directly passed to the emd solver

.. warning::

When dealing with a large number of points, the EMD solver may face

some instabilities, especially when the mass associated to the dummy

point is large. To avoid them, increase the number of dummy points

(allows a smoother repartition of the mass over the points).

Returns

-------

gamma : (dim_a, dim_b) ndarray

Optimal transportation matrix for the given parameters

log : dict

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

Examples

--------

>>> import ot

>>> a = [.1, .2]

>>> b = [.1, .1]

>>> M = [[0., 1.], [2., 3.]]

>>> np.round(partial_wasserstein(a,b,M), 2)

array([[0.1, 0. ],

[0. , 0.1]])

>>> np.round(partial_wasserstein(a,b,M,m=0.1), 2)

array([[0.1, 0. ],

[0. , 0. ]])

References

----------

.. [28] Caffarelli, L. A., & McCann, R. J. (2010) Free boundaries in

optimal transport and Monge-Ampere obstacle problems. Annals of

mathematics, 673-730.

.. [29] Chapel, L., Alaya, M., Gasso, G. (2020). "Partial Optimal

Transport with Applications on Positive-Unlabeled Learning".

NeurIPS.

See Also

--------

ot.partial.partial_wasserstein_lagrange: Partial Wasserstein with

regularization on the marginals

ot.partial.entropic_partial_wasserstein: Partial Wasserstein with a

entropic regularization parameter

"""

dim_a, dim_b = M.shape

if len(a) == 0:

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

if len(b) == 0:

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

if m is None:

return partial_wasserstein_lagrange(a, b, M, log=log, **kwargs)

elif m < 0:

raise ValueError("Problem infeasible. Parameter m should be greater than 0.")

elif m > nx.min(nx.stack((nx.sum(a), nx.sum(b)))):

raise ValueError(

"Problem infeasible. Parameter m should lower or"

" equal than min(|a|_1, |b|_1)."

)

b_extension = nx.ones(nb_dummies, type_as=b) * (nx.sum(a) - m) / nb_dummies

b_extended = nx.concatenate((b, b_extension))

a_extension = nx.ones(nb_dummies, type_as=a) * (nx.sum(b) - m) / nb_dummies

a_extended = nx.concatenate((a, a_extension))

M_extension = nx.ones((nb_dummies, nb_dummies), type_as=M) * nx.max(M) * 2

M_extended = nx.concatenate(

(

nx.concatenate((M, nx.zeros((M.shape[0], M_extension.shape[1]))), axis=1),

nx.concatenate(

(nx.zeros((M_extension.shape[0], M.shape[1])), M_extension), axis=1

),

axis=0,

)

gamma, log_emd = emd(a_extended, b_extended, M_extended, log=True, **kwargs)

gamma = gamma[: len(a), : len(b)]

raise ValueError(

"Error in the EMD resolution: try to increase the number of dummy points"

)

log_emd["partial_w_dist"] = nx.sum(M * gamma)

log_emd["u"] = log_emd["u"][: len(a)]

log_emd["v"] = log_emd["v"][: len(b)]

if log:

return gamma, log_emd

else:

return gamma</div>

def partial_wasserstein2(a, b, M, m=None, nb_dummies=1, log=False, **kwargs):

r"""

Solves the partial optimal transport problem for the quadratic cost

and returns the partial GW discrepancy

The function considers the following problem:

.. math::

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

.. math::

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

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

\gamma &\geq 0

\mathbf{1}^T \gamma^T \mathbf{1} = m &\leq \min\{\|\mathbf{a}\|_1, \|\mathbf{b}\|_1\}

where :

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

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

- `m` is the amount of mass to be transported

Parameters

----------

a : np.ndarray (dim_a,)

Unnormalized histogram of dimension `dim_a`

b : np.ndarray (dim_b,)

Unnormalized histograms of dimension `dim_b`

M : np.ndarray (dim_a, dim_b)

cost matrix for the quadratic cost

m : float, optional

amount of mass to be transported

nb_dummies : int, optional, default:1

number of reservoir points to be added (to avoid numerical

instabilities, increase its value if an error is raised)

log : bool, optional

record log if True

**kwargs : dict

parameters can be directly passed to the emd solver

.. warning::

When dealing with a large number of points, the EMD solver may face

some instabilities, especially when the mass associated to the dummy

point is large. To avoid them, increase the number of dummy points

(allows a smoother repartition of the mass over the points).

Returns

-------

GW: float

partial GW discrepancy

log : dict

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

Examples

--------

>>> import ot

>>> a=[.1, .2]

>>> b=[.1, .1]

>>> M=[[0., 1.], [2., 3.]]

>>> np.round(partial_wasserstein2(a, b, M), 1)

0.3

>>> np.round(partial_wasserstein2(a,b,M,m=0.1), 1)

0.0

References

----------

.. [28] Caffarelli, L. A., & McCann, R. J. (2010) Free boundaries in

optimal transport and Monge-Ampere obstacle problems. Annals of

mathematics, 673-730.

.. [29] Chapel, L., Alaya, M., Gasso, G. (2020). "Partial Optimal

Transport with Applications on Positive-Unlabeled Learning".

NeurIPS.

"""

partial_gw, log_w = partial_wasserstein(a, b, M, m, nb_dummies, log=True, **kwargs)

log_w["T"] = partial_gw

if log:

return nx.sum(partial_gw * M), log_w

else:

def entropic_partial_wasserstein(

a,

b,

M,

reg,

m=None,

method="sinkhorn",

numItermax=1000,

stopThr=1e-100,

verbose=False,

log=False,

):

r"""

Solves the partial optimal transport problem

and returns the OT plan

The function considers the following problem:

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma,

\mathbf{M} \rangle_F + \mathrm{reg} \cdot\Omega(\gamma)

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

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

\gamma &\geq 0 \\

\mathbf{1}^T \gamma^T \mathbf{1} = m

&\leq \min\{\|\mathbf{a}\|_1, \|\mathbf{b}\|_1\} \\

where :

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

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

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

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

- `m` is the amount of mass to be transported

The formulation of the problem has been proposed in

:ref:`[3] <references-entropic-partial-wasserstein>` (prop. 5)

**Choosing a solver**

By default (``method='sinkhorn'``) the problem is solved with the classical

Sinkhorn-like iterations in the multiplicative (exponential) domain. These

are fast, but the kernel :math:`K = e^{-\mathbf{M}/\mathrm{reg}}` underflows

to zero for small ``reg`` (or large cost values), making the plan collapse

to ``NaN``/``Inf`` (see Issue #723). When this happens, set

``method='sinkhorn_log'`` to switch to the mathematically equivalent

log-domain solver :any:`entropic_partial_wasserstein_logscale`, which relies

on the log-sum-exp trick and stays finite at the price of being slower.

Parameters

----------

a : np.ndarray (dim_a,)

Unnormalized histogram of dimension `dim_a`

b : np.ndarray (dim_b,)

Unnormalized histograms of dimension `dim_b`

M : np.ndarray (dim_a, dim_b)

cost matrix

reg : float

Regularization term > 0

m : float, optional

Amount of mass to be transported

method : str, optional

Solver used, either ``'sinkhorn'`` (classical multiplicative-domain

solver, default) or ``'sinkhorn_log'`` (numerically stable log-domain

solver). See the *Choosing a solver* note above.

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) ndarray

Optimal transportation matrix for the given parameters

log : dict

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

Examples

--------

>>> import ot

>>> a = [.1, .2]

>>> b = [.1, .1]

>>> M = [[0., 1.], [2., 3.]]

>>> np.round(entropic_partial_wasserstein(a, b, M, 1, 0.1), 2)

array([[0.06, 0.02],

[0.01, 0. ]])

.. _references-entropic-partial-wasserstein:

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.

See Also

--------

ot.partial.partial_wasserstein: exact Partial Wasserstein

ot.partial.entropic_partial_wasserstein_logscale: stable log-domain solver

"""

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

return _entropic_partial_wasserstein_sinkhorn(

a,

b,

M,

reg,

m=m,

numItermax=numItermax,

stopThr=stopThr,

verbose=verbose,

log=log,

)

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

return entropic_partial_wasserstein_logscale(

a,

b,

M,

reg,

m=m,

numItermax=numItermax,

stopThr=stopThr,

verbose=verbose,

log=log,

)

else:

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

def _entropic_partial_wasserstein_sinkhorn(

a, b, M, reg, m=None, numItermax=1000, stopThr=1e-100, verbose=False, log=False

):

r"""Classical multiplicative-domain solver for the entropic partial OT

problem.

This is the backend used by

``entropic_partial_wasserstein(..., method="sinkhorn")``; see that function

for the problem definition, parameters and references. It is kept as a

separate solver so that the public ``entropic_partial_wasserstein`` is a

thin wrapper dispatching between this solver and

:any:`entropic_partial_wasserstein_logscale`.

"""

dim_a, dim_b = M.shape

dx = nx.ones(dim_a, type_as=a)

dy = nx.ones(dim_b, type_as=b)

if m is None:

m = nx.min(nx.stack((nx.sum(a), nx.sum(b)))) * 1.0

if m < 0:

raise ValueError("Problem infeasible. Parameter m should be greater than 0.")

if m > nx.min(nx.stack((nx.sum(a), nx.sum(b)))):

raise ValueError(

"Problem infeasible. Parameter m should lower or"

" equal than min(|a|_1, |b|_1)."

)

log_e = {"err": []}

if nx.__name__ == "numpy":

# Next 3 lines equivalent to K=nx.exp(-M/reg), but faster to compute

K = np.empty(M.shape, dtype=M.dtype)

np.divide(M, -reg, out=K)

np.exp(K, out=K)

np.multiply(K, m / np.sum(K), out=K)

else:

K = nx.exp(-M / reg)

K = K * m / nx.sum(K)

err, cpt = 1, 0

q1 = nx.ones(K.shape, type_as=K)

q2 = nx.ones(K.shape, type_as=K)

q3 = nx.ones(K.shape, type_as=K)

while err > stopThr and cpt < numItermax:

Kprev = K

K = K * q1

K1 = nx.dot(nx.diag(nx.minimum(a / nx.sum(K, axis=1), dx)), K)

q1 = q1 * Kprev / K1

K1prev = K1

K1 = K1 * q2

K2 = nx.dot(K1, nx.diag(nx.minimum(b / nx.sum(K1, axis=0), dy)))

q2 = q2 * K1prev / K2

K2prev = K2

K2 = K2 * q3

K = K2 * (m / nx.sum(K2))

q3 = q3 * K2prev / K

if nx.any(nx.isnan(K)) or nx.any(nx.isinf(K)):

print("Warning: numerical errors at iteration", cpt)

break

if cpt % 10 == 0:

err = nx.norm(Kprev - K)

if log:

log_e["err"].append(err)

if verbose:

if cpt % 200 == 0:

print("{:5s}|{:12s}".format("It.", "Err") + "\n" + "-" * 11)

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

cpt = cpt + 1

log_e["partial_w_dist"] = nx.sum(M * K)

if log:

return K, log_e

else:

return K

def entropic_partial_wasserstein_logscale(

a, b, M, reg, m=None, numItermax=1000, stopThr=1e-100, verbose=False, log=False

):

r"""

Solves the partial optimal transport problem

and returns the OT plan

This function solves exactly the same problem as

:any:`entropic_partial_wasserstein`, but it is in log-scale, i.e., the log-sum-exp trick is used,

so it is more stable but slower.

The input and output format is the same as :any:`entropic_partial_wasserstein`.

The function considers the following problem:

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma,

\mathbf{M} \rangle_F + \mathrm{reg} \cdot\Omega(\gamma)

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

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

\gamma &\geq 0 \\

\mathbf{1}^T \gamma^T \mathbf{1} = m

&\leq \min\{\|\mathbf{a}\|_1, \|\mathbf{b}\|_1\} \\

where :

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

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

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

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

- `m` is the amount of mass to be transported

The formulation of the problem has been proposed in

:ref:`[3] <references-entropic-partial-wasserstein>` (prop. 5)

Parameters

----------

a : np.ndarray (dim_a,)

Unnormalized histogram of dimension `dim_a`

b : np.ndarray (dim_b,)

Unnormalized histograms of dimension `dim_b`

M : np.ndarray (dim_a, dim_b)

cost matrix

reg : float

Regularization term > 0

m : float, optional

Amount of mass to be transported

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) ndarray

Optimal transportation matrix for the given parameters

log : dict

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

Examples

--------

>>> import ot

>>> a = [.1, .2]

>>> b = [.1, .1]

>>> M = [[0., 1.], [2., 3.]]

>>> np.round(entropic_partial_wasserstein_logscale(a, b, M, 1, 0.1), 2)

array([[0.06, 0.02],

[0.01, 0. ]])

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.

See Also

--------

ot.partial.partial_wasserstein: exact Partial Wasserstein

ot.partial.entropic_partial_wasserstein: classical entropic solver and

dispatcher (this solver is also reachable through its ``method='sinkhorn_log'`` argument)

"""

dim_a, dim_b = M.shape

Ldx = nx.zeros(dim_a, type_as=a)

Ldy = nx.zeros(dim_b, type_as=b)

La = nx.log(a)

Lb = nx.log(b)

if m is None:

if m < 0:

raise ValueError("Problem infeasible. Parameter m should be greater than 0.")

raise ValueError(

"Problem infeasible. Parameter m should lower or"

" equal than min(|a|_1, |b|_1)."

)

log_e = {"err": []}

LK = -M / reg

LK = LK + nx.log(m) - nx.logsumexp(LK)

err, cpt = 1, 0

Lq1 = nx.zeros(M.shape, type_as=M)

Lq2 = nx.zeros(M.shape, type_as=M)

Lq3 = nx.zeros(M.shape, type_as=M)

LKprev = LK

LK = LK + Lq1

LK1 = nx.reshape(nx.minimum(La - nx.logsumexp(LK, 1), Ldx), (-1, 1)) + LK

Lq1 = Lq1 + LKprev - LK1

LK1prev = LK1

LK1 = LK1 + Lq2

LK2 = LK1 + nx.reshape(nx.minimum(Lb - nx.logsumexp(LK1, 0), Ldy), (1, -1))

Lq2 = Lq2 + LK1prev - LK2

LK2prev = LK2

LK2 = LK2 + Lq3

LK = LK2 + nx.log(m) - nx.logsumexp(LK2)

Lq3 = Lq3 + LK2prev - LK

if nx.any(nx.isnan(LK)) or nx.any(nx.isinf(LK)):

warnings.warn(

f"Numerical errors at iteration {cpt} of entropic_partial_wasserstein_logscale",

stacklevel=2,

)

break

if cpt % 10 == 0:

err = nx.norm(LKprev - LK)

if log:

if verbose:

if cpt % 200 == 0:

cpt = cpt + 1

if log:

return nx.exp(LK), log_e

else:

return nx.exp(LK)</div>

def gwgrad_partial(C1, C2, T):

"""Compute the GW gradient. Note: we can not use the trick in :ref:`[12] <references-gwgrad-partial>`

as the marginals may not sum to 1.

.. note:: This function will be deprecated in a near future, please use

`ot.gromov.gwggrad` instead.

Parameters

----------

C1: array of shape (n_p,n_p)

intra-source (P) cost matrix

C2: array of shape (n_u,n_u)

intra-target (U) cost matrix

T : array of shape(n_p+nb_dummies, n_u) (default: None)

Transport matrix

Returns

-------

numpy.array of shape (n_p+nb_dummies, n_u)

gradient

.. _references-gwgrad-partial:

References

----------

.. [12] Peyré, Gabriel, Marco Cuturi, and Justin Solomon,

"Gromov-Wasserstein averaging of kernel and distance matrices."

International Conference on Machine Learning (ICML). 2016.

"""

warnings.warn(

"This function will be deprecated in a near future, please use "

"ot.gromov.gwggrad` instead.",

stacklevel=2,

)

cC1 = np.dot(C1**2 / 2, np.dot(T, np.ones(C2.shape[0]).reshape(-1, 1)))

cC2 = np.dot(np.dot(np.ones(C1.shape[0]).reshape(1, -1), T), C2**2 / 2)

constC = cC1 + cC2

A = -np.dot(C1, T).dot(C2.T)

tens = constC + A

return tens * 2</div>

def gwloss_partial(C1, C2, T):

"""Compute the GW loss.

.. note:: This function will be deprecated in a near future, please use

`ot.gromov.gwloss` instead.

Parameters

----------

C1: array of shape (n_p,n_p)

intra-source (P) cost matrix

C2: array of shape (n_u,n_u)

intra-target (U) cost matrix

T : array of shape(n_p+nb_dummies, n_u) (default: None)

Transport matrix

Returns

-------

GW loss

"""

warnings.warn(

"This function will be deprecated in a near future, please use "

"ot.gromov.gwloss` instead.",

stacklevel=2,

)

g = gwgrad_partial(C1, C2, T) * 0.5

return np.sum(g * T)</div>

def partial_gromov_wasserstein(

C1,

C2,

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