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# -*- coding: utf-8 -*-
"""
Semi-relaxed Gromov-Wasserstein and Fused-Gromov-Wasserstein solvers.
"""
# Author: Rémi Flamary <remi.flamary@unice.fr>
# Cédric Vincent-Cuaz <cedvincentcuaz@gmail.com>
#
# License: MIT License
import numpy as np
from ..utils import list_to_array, unif
from ..optim import semirelaxed_cg, solve_1d_linesearch_quad
from ..backend import get_backend
from ._utils import init_matrix_semirelaxed, gwloss, gwggrad
def semirelaxed_gromov_wasserstein(C1, C2, p, loss_fun='square_loss', symmetric=None, log=False, G0=None,
max_iter=1e4, tol_rel=1e-9, tol_abs=1e-9, **kwargs):
r"""
Returns the semi-relaxed gromov-wasserstein divergence transport from :math:`(\mathbf{C_1}, \mathbf{p})` to :math:`\mathbf{C_2}`
The function solves the following optimization problem:
.. math::
\mathbf{srGW} = \mathop{\arg \min}_\mathbf{T} \quad \sum_{i,j,k,l}
L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j} \mathbf{T}_{k,l}
s.t. \ \mathbf{\gamma} \mathbf{1} &= \mathbf{p}
\mathbf{\gamma} &\geq 0
Where :
- :math:`\mathbf{C_1}`: Metric cost matrix in the source space
- :math:`\mathbf{C_2}`: Metric cost matrix in the target space
- :math:`\mathbf{p}`: distribution in the source space
- `L`: loss function to account for the misfit between the similarity matrices
.. note:: This function is backend-compatible and will work on arrays
from all compatible backends. However all the steps in the conditional
gradient are not differentiable.
Parameters
----------
C1 : array-like, shape (ns, ns)
Metric cost matrix in the source space
C2 : array-like, shape (nt, nt)
Metric cost matrix in the target space
p : array-like, shape (ns,)
Distribution in the source space
loss_fun : str
loss function used for the solver either 'square_loss' or 'kl_loss'.
'kl_loss' is not implemented yet and will raise an error.
symmetric : bool, optional
Either C1 and C2 are to be assumed symmetric or not.
If let to its default None value, a symmetry test will be conducted.
Else if set to True (resp. False), C1 and C2 will be assumed symmetric (resp. asymetric).
verbose : bool, optional
Print information along iterations
log : bool, optional
record log if True
G0: array-like, shape (ns,nt), optional
If None the initial transport plan of the solver is pq^T.
Otherwise G0 must satisfy marginal constraints and will be used as initial transport of the solver.
max_iter : int, optional
Max number of iterations
tol_rel : float, optional
Stop threshold on relative error (>0)
tol_abs : float, optional
Stop threshold on absolute error (>0)
**kwargs : dict
parameters can be directly passed to the ot.optim.cg solver
Returns
-------
T : array-like, shape (`ns`, `nt`)
Coupling between the two spaces that minimizes:
:math:`\sum_{i,j,k,l} L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j} \mathbf{T}_{k,l}`
log : dict
Convergence information and loss.
References
----------
.. [48] Cédric Vincent-Cuaz, Rémi Flamary, Marco Corneli, Titouan Vayer, Nicolas Courty.
"Semi-relaxed Gromov-Wasserstein divergence and applications on graphs"
International Conference on Learning Representations (ICLR), 2022.
"""
if loss_fun == 'kl_loss':
raise NotImplementedError()
p = list_to_array(p)
if G0 is None:
nx = get_backend(p, C1, C2)
else:
nx = get_backend(p, C1, C2, G0)
if symmetric is None:
symmetric = nx.allclose(C1, C1.T, atol=1e-10) and nx.allclose(C2, C2.T, atol=1e-10)
if G0 is None:
q = unif(C2.shape[0], type_as=p)
G0 = nx.outer(p, q)
else:
q = nx.sum(G0, 0)
# Check first marginal of G0
np.testing.assert_allclose(nx.sum(G0, 1), p, atol=1e-08)
constC, hC1, hC2, fC2t = init_matrix_semirelaxed(C1, C2, p, loss_fun, nx)
ones_p = nx.ones(p.shape[0], type_as=p)
def f(G):
qG = nx.sum(G, 0)
marginal_product = nx.outer(ones_p, nx.dot(qG, fC2t))
return gwloss(constC + marginal_product, hC1, hC2, G, nx)
if symmetric:
def df(G):
qG = nx.sum(G, 0)
marginal_product = nx.outer(ones_p, nx.dot(qG, fC2t))
return gwggrad(constC + marginal_product, hC1, hC2, G, nx)
else:
constCt, hC1t, hC2t, fC2 = init_matrix_semirelaxed(C1.T, C2.T, p, loss_fun, nx)
def df(G):
qG = nx.sum(G, 0)
marginal_product_1 = nx.outer(ones_p, nx.dot(qG, fC2t))
marginal_product_2 = nx.outer(ones_p, nx.dot(qG, fC2))
return 0.5 * (gwggrad(constC + marginal_product_1, hC1, hC2, G, nx) + gwggrad(constCt + marginal_product_2, hC1t, hC2t, G, nx))
def line_search(cost, G, deltaG, Mi, cost_G, **kwargs):
return solve_semirelaxed_gromov_linesearch(G, deltaG, cost_G, C1, C2, ones_p, M=0., reg=1., nx=nx, **kwargs)
if log:
res, log = semirelaxed_cg(p, q, 0., 1., f, df, G0, line_search, log=True, numItermax=max_iter, stopThr=tol_rel, stopThr2=tol_abs, **kwargs)
log['srgw_dist'] = log['loss'][-1]
return res, log
else:
return semirelaxed_cg(p, q, 0., 1., f, df, G0, line_search, log=False, numItermax=max_iter, stopThr=tol_rel, stopThr2=tol_abs, **kwargs)
def semirelaxed_gromov_wasserstein2(C1, C2, p, loss_fun='square_loss', symmetric=None, log=False, G0=None,
max_iter=1e4, tol_rel=1e-9, tol_abs=1e-9, **kwargs):
r"""
Returns the semi-relaxed gromov-wasserstein divergence from :math:`(\mathbf{C_1}, \mathbf{p})` to :math:`\mathbf{C_2}`
The function solves the following optimization problem:
.. math::
srGW = \min_\mathbf{T} \quad \sum_{i,j,k,l}
L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j} \mathbf{T}_{k,l}
s.t. \ \mathbf{\gamma} \mathbf{1} &= \mathbf{p}
\mathbf{\gamma} &\geq 0
Where :
- :math:`\mathbf{C_1}`: Metric cost matrix in the source space
- :math:`\mathbf{C_2}`: Metric cost matrix in the target space
- :math:`\mathbf{p}`: distribution in the source space
- `L`: loss function to account for the misfit between the similarity
matrices
Note that when using backends, this loss function is differentiable wrt the
matrices (C1, C2) but not yet for the weights p.
.. note:: This function is backend-compatible and will work on arrays
from all compatible backends. However all the steps in the conditional
gradient are not differentiable.
Parameters
----------
C1 : array-like, shape (ns, ns)
Metric cost matrix in the source space
C2 : array-like, shape (nt, nt)
Metric cost matrix in the target space
p : array-like, shape (ns,)
Distribution in the source space.
loss_fun : str
loss function used for the solver either 'square_loss' or 'kl_loss'.
'kl_loss' is not implemented yet and will raise an error.
symmetric : bool, optional
Either C1 and C2 are to be assumed symmetric or not.
If let to its default None value, a symmetry test will be conducted.
Else if set to True (resp. False), C1 and C2 will be assumed symmetric (resp. asymetric).
verbose : bool, optional
Print information along iterations
log : bool, optional
record log if True
G0: array-like, shape (ns,nt), optional
If None the initial transport plan of the solver is pq^T.
Otherwise G0 must satisfy marginal constraints and will be used as initial transport of the solver.
max_iter : int, optional
Max number of iterations
tol_rel : float, optional
Stop threshold on relative error (>0)
tol_abs : float, optional
Stop threshold on absolute error (>0)
**kwargs : dict
parameters can be directly passed to the ot.optim.cg solver
Returns
-------
srgw : float
Semi-relaxed Gromov-Wasserstein divergence
log : dict
convergence information and Coupling matrix
References
----------
.. [48] Cédric Vincent-Cuaz, Rémi Flamary, Marco Corneli, Titouan Vayer, Nicolas Courty.
"Semi-relaxed Gromov-Wasserstein divergence and applications on graphs"
International Conference on Learning Representations (ICLR), 2022.
"""
nx = get_backend(p, C1, C2)
T, log_srgw = semirelaxed_gromov_wasserstein(
C1, C2, p, loss_fun, symmetric, log=True, G0=G0,
max_iter=max_iter, tol_rel=tol_rel, tol_abs=tol_abs, **kwargs)
q = nx.sum(T, 0)
log_srgw['T'] = T
srgw = log_srgw['srgw_dist']
if loss_fun == 'square_loss':
gC1 = 2 * C1 * nx.outer(p, p) - 2 * nx.dot(T, nx.dot(C2, T.T))
gC2 = 2 * C2 * nx.outer(q, q) - 2 * nx.dot(T.T, nx.dot(C1, T))
srgw = nx.set_gradients(srgw, (C1, C2), (gC1, gC2))
if log:
return srgw, log_srgw
else:
return srgw
def semirelaxed_fused_gromov_wasserstein(M, C1, C2, p, loss_fun='square_loss', symmetric=None, alpha=0.5, G0=None, log=False,
max_iter=1e4, tol_rel=1e-9, tol_abs=1e-9, **kwargs):
r"""
Computes the semi-relaxed FGW transport between two graphs (see :ref:`[48] <references-semirelaxed-fused-gromov-wasserstein>`)
.. math::
\gamma = \mathop{\arg \min}_\gamma \quad (1 - \alpha) \langle \gamma, \mathbf{M} \rangle_F +
\alpha \sum_{i,j,k,l} L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j} \mathbf{T}_{k,l}
s.t. \ \mathbf{\gamma} \mathbf{1} &= \mathbf{p}
\mathbf{\gamma} &\geq 0
where :
- :math:`\mathbf{M}` is the (`ns`, `nt`) metric cost matrix
- :math:`\mathbf{p}` source weights (sum to 1)
- `L` is a loss function to account for the misfit between the similarity matrices
.. note:: This function is backend-compatible and will work on arrays
from all compatible backends. However all the steps in the conditional
gradient are not differentiable.
The algorithm used for solving the problem is conditional gradient as discussed in :ref:`[48] <references-semirelaxed-fused-gromov-wasserstein>`
Parameters
----------
M : array-like, shape (ns, nt)
Metric cost matrix between features across domains
C1 : array-like, shape (ns, ns)
Metric cost matrix representative of the structure in the source space
C2 : array-like, shape (nt, nt)
Metric cost matrix representative of the structure in the target space
p : array-like, shape (ns,)
Distribution in the source space
loss_fun : str
loss function used for the solver either 'square_loss' or 'kl_loss'.
'kl_loss' is not implemented yet and will raise an error.
symmetric : bool, optional
Either C1 and C2 are to be assumed symmetric or not.
If let to its default None value, a symmetry test will be conducted.
Else if set to True (resp. False), C1 and C2 will be assumed symmetric (resp. asymetric).
alpha : float, optional
Trade-off parameter (0 < alpha < 1)
G0: array-like, shape (ns,nt), optional
If None the initial transport plan of the solver is pq^T.
Otherwise G0 must satisfy marginal constraints and will be used as initial transport of the solver.
log : bool, optional
record log if True
max_iter : int, optional
Max number of iterations
tol_rel : float, optional
Stop threshold on relative error (>0)
tol_abs : float, optional
Stop threshold on absolute error (>0)
**kwargs : dict
parameters can be directly passed to the ot.optim.cg solver
Returns
-------
gamma : array-like, shape (`ns`, `nt`)
Optimal transportation matrix for the given parameters.
log : dict
Log dictionary return only if log==True in parameters.
.. _references-semirelaxed-fused-gromov-wasserstein:
References
----------
.. [24] Vayer Titouan, Chapel Laetitia, Flamary Rémi, Tavenard Romain
and Courty Nicolas "Optimal Transport for structured data with
application on graphs", International Conference on Machine Learning
(ICML). 2019.
.. [48] Cédric Vincent-Cuaz, Rémi Flamary, Marco Corneli, Titouan Vayer, Nicolas Courty.
"Semi-relaxed Gromov-Wasserstein divergence and applications on graphs"
International Conference on Learning Representations (ICLR), 2022.
"""
if loss_fun == 'kl_loss':
raise NotImplementedError()
p = list_to_array(p)
if G0 is None:
nx = get_backend(p, C1, C2, M)
else:
nx = get_backend(p, C1, C2, M, G0)
if symmetric is None:
symmetric = nx.allclose(C1, C1.T, atol=1e-10) and nx.allclose(C2, C2.T, atol=1e-10)
if G0 is None:
q = unif(C2.shape[0], type_as=p)
G0 = nx.outer(p, q)
else:
q = nx.sum(G0, 0)
# Check marginals of G0
np.testing.assert_allclose(nx.sum(G0, 1), p, atol=1e-08)
constC, hC1, hC2, fC2t = init_matrix_semirelaxed(C1, C2, p, loss_fun, nx)
ones_p = nx.ones(p.shape[0], type_as=p)
def f(G):
qG = nx.sum(G, 0)
marginal_product = nx.outer(ones_p, nx.dot(qG, fC2t))
return gwloss(constC + marginal_product, hC1, hC2, G, nx)
if symmetric:
def df(G):
qG = nx.sum(G, 0)
marginal_product = nx.outer(ones_p, nx.dot(qG, fC2t))
return gwggrad(constC + marginal_product, hC1, hC2, G, nx)
else:
constCt, hC1t, hC2t, fC2 = init_matrix_semirelaxed(C1.T, C2.T, p, loss_fun, nx)
def df(G):
qG = nx.sum(G, 0)
marginal_product_1 = nx.outer(ones_p, nx.dot(qG, fC2t))
marginal_product_2 = nx.outer(ones_p, nx.dot(qG, fC2))
return 0.5 * (gwggrad(constC + marginal_product_1, hC1, hC2, G, nx) + gwggrad(constCt + marginal_product_2, hC1t, hC2t, G, nx))
def line_search(cost, G, deltaG, Mi, cost_G, **kwargs):
return solve_semirelaxed_gromov_linesearch(
G, deltaG, cost_G, C1, C2, ones_p, M=(1 - alpha) * M, reg=alpha, nx=nx, **kwargs)
if log:
res, log = semirelaxed_cg(p, q, (1 - alpha) * M, alpha, f, df, G0, line_search, log=True, numItermax=max_iter, stopThr=tol_rel, stopThr2=tol_abs, **kwargs)
log['srfgw_dist'] = log['loss'][-1]
return res, log
else:
return semirelaxed_cg(p, q, (1 - alpha) * M, alpha, f, df, G0, line_search, log=False, numItermax=max_iter, stopThr=tol_rel, stopThr2=tol_abs, **kwargs)
def semirelaxed_fused_gromov_wasserstein2(M, C1, C2, p, loss_fun='square_loss', symmetric=None, alpha=0.5, G0=None, log=False,
max_iter=1e4, tol_rel=1e-9, tol_abs=1e-9, **kwargs):
r"""
Computes the semi-relaxed FGW divergence between two graphs (see :ref:`[48] <references-semirelaxed-fused-gromov-wasserstein2>`)
.. math::
\min_\gamma \quad (1 - \alpha) \langle \gamma, \mathbf{M} \rangle_F + \alpha \sum_{i,j,k,l}
L(\mathbf{C_1}_{i,k}, \mathbf{C_2}_{j,l}) \mathbf{T}_{i,j} \mathbf{T}_{k,l}
s.t. \ \mathbf{\gamma} \mathbf{1} &= \mathbf{p}
\mathbf{\gamma} &\geq 0
where :
- :math:`\mathbf{M}` is the (`ns`, `nt`) metric cost matrix
- :math:`\mathbf{p}` source weights (sum to 1)
- `L` is a loss function to account for the misfit between the similarity matrices
The algorithm used for solving the problem is conditional gradient as
discussed in :ref:`[48] <semirelaxed-fused-gromov-wasserstein2>`
Note that when using backends, this loss function is differentiable wrt the
matrices (C1, C2) but not yet for the weights p.
.. note:: This function is backend-compatible and will work on arrays
from all compatible backends. However all the steps in the conditional
gradient are not differentiable.
Parameters
----------
M : array-like, shape (ns, nt)
Metric cost matrix between features across domains
C1 : array-like, shape (ns, ns)
Metric cost matrix representative of the structure in the source space.
C2 : array-like, shape (nt, nt)
Metric cost matrix representative of the structure in the target space.
p : array-like, shape (ns,)
Distribution in the source space.
loss_fun : str, optional
loss function used for the solver either 'square_loss' or 'kl_loss'.
'kl_loss' is not implemented yet and will raise an error.
symmetric : bool, optional
Either C1 and C2 are to be assumed symmetric or not.
If let to its default None value, a symmetry test will be conducted.
Else if set to True (resp. False), C1 and C2 will be assumed symmetric (resp. asymetric).
alpha : float, optional
Trade-off parameter (0 < alpha < 1)
G0: array-like, shape (ns,nt), optional
If None the initial transport plan of the solver is pq^T.
Otherwise G0 must satisfy marginal constraints and will be used as initial transport of the solver.
log : bool, optional
Record log if True.
max_iter : int, optional
Max number of iterations
tol_rel : float, optional
Stop threshold on relative error (>0)
tol_abs : float, optional
Stop threshold on absolute error (>0)
**kwargs : dict
Parameters can be directly passed to the ot.optim.cg solver.
Returns
-------
srfgw-divergence : float
Semi-relaxed Fused gromov wasserstein divergence for the given parameters.
log : dict
Log dictionary return only if log==True in parameters.
.. _references-semirelaxed-fused-gromov-wasserstein2:
References
----------
.. [24] Vayer Titouan, Chapel Laetitia, Flamary Rémi, Tavenard Romain
and Courty Nicolas "Optimal Transport for structured data with
application on graphs", International Conference on Machine Learning
(ICML). 2019.
.. [48] Cédric Vincent-Cuaz, Rémi Flamary, Marco Corneli, Titouan Vayer, Nicolas Courty.
"Semi-relaxed Gromov-Wasserstein divergence and applications on graphs"
International Conference on Learning Representations (ICLR), 2022.
"""
nx = get_backend(p, C1, C2, M)
T, log_fgw = semirelaxed_fused_gromov_wasserstein(
M, C1, C2, p, loss_fun, symmetric, alpha, G0, log=True,
max_iter=max_iter, tol_rel=tol_rel, tol_abs=tol_abs, **kwargs)
q = nx.sum(T, 0)
srfgw_dist = log_fgw['srfgw_dist']
log_fgw['T'] = T
if loss_fun == 'square_loss':
gC1 = 2 * C1 * nx.outer(p, p) - 2 * nx.dot(T, nx.dot(C2, T.T))
gC2 = 2 * C2 * nx.outer(q, q) - 2 * nx.dot(T.T, nx.dot(C1, T))
srfgw_dist = nx.set_gradients(srfgw_dist, (C1, C2, M),
(alpha * gC1, alpha * gC2, (1 - alpha) * T))
if log:
return srfgw_dist, log_fgw
else:
return srfgw_dist
def solve_semirelaxed_gromov_linesearch(G, deltaG, cost_G, C1, C2, ones_p,
M, reg, alpha_min=None, alpha_max=None, nx=None, **kwargs):
"""
Solve the linesearch in the FW iterations
Parameters
----------
G : array-like, shape(ns,nt)
The transport map at a given iteration of the FW
deltaG : array-like (ns,nt)
Difference between the optimal map found by linearization in the FW algorithm and the value at a given iteration
cost_G : float
Value of the cost at `G`
C1 : array-like (ns,ns)
Structure matrix in the source domain.
C2 : array-like (nt,nt)
Structure matrix in the target domain.
ones_p: array-like (ns,1)
Array of ones of size ns
M : array-like (ns,nt)
Cost matrix between the features.
reg : float
Regularization parameter.
alpha_min : float, optional
Minimum value for alpha
alpha_max : float, optional
Maximum value for alpha
nx : backend, optional
If let to its default value None, a backend test will be conducted.
Returns
-------
alpha : float
The optimal step size of the FW
fc : int
nb of function call. Useless here
cost_G : float
The value of the cost for the next iteration
References
----------
.. [48] Cédric Vincent-Cuaz, Rémi Flamary, Marco Corneli, Titouan Vayer, Nicolas Courty.
"Semi-relaxed Gromov-Wasserstein divergence and applications on graphs"
International Conference on Learning Representations (ICLR), 2021.
"""
if nx is None:
G, deltaG, C1, C2, M = list_to_array(G, deltaG, C1, C2, M)
if isinstance(M, int) or isinstance(M, float):
nx = get_backend(G, deltaG, C1, C2)
else:
nx = get_backend(G, deltaG, C1, C2, M)
qG, qdeltaG = nx.sum(G, 0), nx.sum(deltaG, 0)
dot = nx.dot(nx.dot(C1, deltaG), C2.T)
C2t_square = C2.T ** 2
dot_qG = nx.dot(nx.outer(ones_p, qG), C2t_square)
dot_qdeltaG = nx.dot(nx.outer(ones_p, qdeltaG), C2t_square)
a = reg * nx.sum((dot_qdeltaG - 2 * dot) * deltaG)
b = nx.sum(M * deltaG) + reg * (nx.sum((dot_qdeltaG - 2 * dot) * G) + nx.sum((dot_qG - 2 * nx.dot(nx.dot(C1, G), C2.T)) * deltaG))
alpha = solve_1d_linesearch_quad(a, b)
if alpha_min is not None or alpha_max is not None:
alpha = np.clip(alpha, alpha_min, alpha_max)
# the new cost can be deduced from the line search quadratic function
cost_G = cost_G + a * (alpha ** 2) + b * alpha
return alpha, 1, cost_G
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