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<!DOCTYPE html>

<head>

<title>ot.partial — POT Python Optimal Transport 0.9.6dev0 documentation</title>

</head>

POT Python Optimal Transport

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<li class="breadcrumb-item"><a href="../index.html">Module code</a></li>

<li class="breadcrumb-item active">ot.partial</li>

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<h1>Source code for ot.partial</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

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, 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)

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(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

"""

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)

cpt = cpt + 1

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

if log:

return K, log_e

else:

return K</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,

p,

q,

m=None,

nb_dummies=1,

G0=None,

thres=1,

numItermax=1000,

tol=1e-7,

log=False,

verbose=False,

**kwargs,

):

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

.. 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:`\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 the sample weights

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

The formulation of the problem has been proposed in

:ref:`[29] <references-partial-gromov-wasserstein>`

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

`ot.gromov.partial_gromov_wasserstein` instead.

Parameters

----------

C1 : ndarray, shape (ns, ns)

Metric cost matrix in the source space

C2 : ndarray, shape (nt, nt)

Metric costfr matrix in the target space

p : ndarray, shape (ns,)

Distribution in the source space

q : ndarray, shape (nt,)

Distribution in the target space

m : float, optional

Amount of mass to be transported

(default: :math:`\min\{\|\mathbf{p}\|_1, \|\mathbf{q}\|_1\}`)

nb_dummies : int, optional

Number of dummy points to add (avoid instabilities in the EMD solver)

G0 : ndarray, shape (ns, nt), optional

Initialization of the transportation matrix

thres : float, optional

quantile of the gradient matrix to populate the cost matrix when 0

(default: 1)

numItermax : int, optional

Max number of iterations

tol : float, optional

tolerance for stopping iterations

log : bool, optional

return log if True

verbose : bool, optional

Print information along iterations

**kwargs : dict

parameters can be directly passed to the emd solver

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

>>> import scipy as sp

>>> a = np.array([0.25] * 4)

>>> b = np.array([0.25] * 4)

>>> x = np.array([1,2,100,200]).reshape((-1,1))

>>> y = np.array([3,2,98,199]).reshape((-1,1))

>>> C1 = sp.spatial.distance.cdist(x, x)

>>> C2 = sp.spatial.distance.cdist(y, y)

>>> np.round(partial_gromov_wasserstein(C1, C2, a, b),2)

array([[0. , 0.25, 0. , 0. ],

[0.25, 0. , 0. , 0. ],

[0. , 0. , 0.25, 0. ],

[0. , 0. , 0. , 0.25]])

>>> np.round(partial_gromov_wasserstein(C1, C2, a, b, m=0.25),2)

array([[0. , 0. , 0. , 0. ],

[0. , 0. , 0. , 0. ],

[0. , 0. , 0.25, 0. ],

[0. , 0. , 0. , 0. ]])

.. _references-partial-gromov-wasserstein:

References

----------

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

Transport with Applications on Positive-Unlabeled Learning".

NeurIPS.

"""

warnings.warn(

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

"ot.gromov.partial_gromov_wasserstein` instead.",

stacklevel=2,

)

if m is None:

m = np.min((np.sum(p), np.sum(q)))

elif m < 0:

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

elif m > np.min((np.sum(p), np.sum(q))):

raise ValueError(

"Problem infeasible. Parameter m should lower or"

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

)

if G0 is None:

G0 = (

np.outer(p, q) * m / (np.sum(p) * np.sum(q))

) # make sure |G0|=m, G01_m\leq p, G0.T1_n\leq q.

dim_G_extended = (len(p) + nb_dummies, len(q) + nb_dummies)

q_extended = np.append(q, [(np.sum(p) - m) / nb_dummies] * nb_dummies)

p_extended = np.append(p, [(np.sum(q) - m) / nb_dummies] * nb_dummies)

cpt = 0

err = 1

if log:

log = {"err": []}

while err > tol and cpt < numItermax:

Gprev = np.copy(G0)

M = 0.5 * gwgrad_partial(

C1, C2, G0

) # rescaling the gradient with 0.5 for line-search while not changing Gc

M_emd = np.zeros(dim_G_extended)

M_emd[: len(p), : len(q)] = M

M_emd[-nb_dummies:, -nb_dummies:] = np.max(M) * 1e2

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

Gc, logemd = emd(p_extended, q_extended, M_emd, log=True, **kwargs)

if logemd["warning"] is not None:

raise ValueError(

"Error in the EMD resolution: try to increase the"

" number of dummy points"

)

G0 = Gc[: len(p), : len(q)]

if cpt % 10 == 0: # to speed up the computations

err = np.linalg.norm(G0 - Gprev)

if log:

log["err"].append(err)

if verbose:

if cpt % 200 == 0:

print(

"{:5s}|{:12s}|{:12s}".format("It.", "Err", "Loss")

+ "\n"

+ "-" * 31

)

print("{:5d}|{:8e}|{:8e}".format(cpt, err, gwloss_partial(C1, C2, G0)))

deltaG = G0 - Gprev

a = gwloss_partial(C1, C2, deltaG)

b = 2 * np.sum(M * deltaG)

if b > 0: # due to numerical precision

gamma = 0

cpt = numItermax

elif a > 0:

gamma = min(1, np.divide(-b, 2.0 * a))

else:

if (a + b) < 0:

gamma = 1

else:

gamma = 0

cpt = numItermax

G0 = Gprev + gamma * deltaG

cpt += 1

if log:

log["partial_gw_dist"] = gwloss_partial(C1, C2, G0)

return G0[: len(p), : len(q)], log

else:

def partial_gromov_wasserstein2(

C1,

C2,

p,

q,

m=None,

nb_dummies=1,

G0=None,

thres=1,

numItermax=1000,

tol=1e-7,

log=False,

verbose=False,

**kwargs,

):

r"""

Solves the partial optimal transport problem

and returns the partial Gromov-Wasserstein discrepancy

The function considers the following problem:

.. math::

GW = \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

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