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# -*- coding: utf-8 -*-
"""
Bregman projections solvers for entropic Gromov-Wasserstein
"""
# Author: Erwan Vautier <erwan.vautier@gmail.com>
# Nicolas Courty <ncourty@irisa.fr>
# Rémi Flamary <remi.flamary@unice.fr>
# Titouan Vayer <titouan.vayer@irisa.fr>
# Cédric Vincent-Cuaz <cedvincentcuaz@gmail.com>
#
# License: MIT License
import numpy as np
from ..bregman import sinkhorn
from ..utils import dist, list_to_array, check_random_state
from ..backend import get_backend
from ._utils import init_matrix, gwloss, gwggrad
from ._utils import update_square_loss, update_kl_loss
def entropic_gromov_wasserstein(C1, C2, p, q, loss_fun, epsilon, symmetric=None, G0=None,
max_iter=1000, tol=1e-9, verbose=False, log=False):
r"""
Returns the gromov-wasserstein transport between :math:`(\mathbf{C_1}, \mathbf{p})` and :math:`(\mathbf{C_2}, \mathbf{q})`
The function solves the following optimization problem:
.. math::
\mathbf{GW} = \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} - \epsilon(H(\mathbf{T}))
s.t. \ \mathbf{T} \mathbf{1} &= \mathbf{p}
\mathbf{T}^T \mathbf{1} &= \mathbf{q}
\mathbf{T} &\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
- :math:`\mathbf{q}`: distribution in the target space
- `L`: loss function to account for the misfit between the similarity matrices
- `H`: entropy
.. note:: If the inner solver `ot.sinkhorn` did not convergence, the
optimal coupling :math:`\mathbf{T}` returned by this function does not
necessarily satisfy the marginal constraints
:math:`\mathbf{T}\mathbf{1}=\mathbf{p}` and
:math:`\mathbf{T}^T\mathbf{1}=\mathbf{q}`. So the returned
Gromov-Wasserstein loss does not necessarily satisfy distance
properties and may be negative.
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
q : array-like, shape (nt,)
Distribution in the target space
loss_fun : string
Loss function used for the solver either 'square_loss' or 'kl_loss'
epsilon : float
Regularization term >0
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).
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 : float, optional
Stop threshold on error (>0)
verbose : bool, optional
Print information along iterations
log : bool, optional
Record log if True.
Returns
-------
T : array-like, shape (`ns`, `nt`)
Optimal coupling between the two spaces
References
----------
.. [12] Gabriel Peyré, Marco Cuturi, and Justin Solomon,
"Gromov-Wasserstein averaging of kernel and distance matrices."
International Conference on Machine Learning (ICML). 2016.
.. [47] Chowdhury, S., & Mémoli, F. (2019). The gromov–wasserstein
distance between networks and stable network invariants.
Information and Inference: A Journal of the IMA, 8(4), 757-787.
"""
C1, C2, p, q = list_to_array(C1, C2, p, q)
if G0 is None:
nx = get_backend(p, q, C1, C2)
G0 = nx.outer(p, q)
else:
nx = get_backend(p, q, C1, C2, G0)
T = G0
constC, hC1, hC2 = init_matrix(C1, C2, p, q, loss_fun, nx)
if symmetric is None:
symmetric = np.allclose(C1, C1.T, atol=1e-10) and np.allclose(C2, C2.T, atol=1e-10)
if not symmetric:
constCt, hC1t, hC2t = init_matrix(C1.T, C2.T, p, q, loss_fun, nx)
cpt = 0
err = 1
if log:
log = {'err': []}
while (err > tol and cpt < max_iter):
Tprev = T
# compute the gradient
if symmetric:
tens = gwggrad(constC, hC1, hC2, T, nx)
else:
tens = 0.5 * (gwggrad(constC, hC1, hC2, T, nx) + gwggrad(constCt, hC1t, hC2t, T, nx))
T = sinkhorn(p, q, tens, epsilon, method='sinkhorn')
if cpt % 10 == 0:
# we can speed up the process by checking for the error only all
# the 10th iterations
err = nx.norm(T - Tprev)
if log:
log['err'].append(err)
if verbose:
if cpt % 200 == 0:
print('{:5s}|{:12s}'.format(
'It.', 'Err') + '\n' + '-' * 19)
print('{:5d}|{:8e}|'.format(cpt, err))
cpt += 1
if log:
log['gw_dist'] = gwloss(constC, hC1, hC2, T, nx)
return T, log
else:
return T
def entropic_gromov_wasserstein2(C1, C2, p, q, loss_fun, epsilon, symmetric=None, G0=None,
max_iter=1000, tol=1e-9, verbose=False, log=False):
r"""
Returns the entropic gromov-wasserstein discrepancy between the two measured similarity matrices :math:`(\mathbf{C_1}, \mathbf{p})` and :math:`(\mathbf{C_2}, \mathbf{q})`
The function solves the following optimization problem:
.. math::
GW = \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} - \epsilon(H(\mathbf{T}))
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
- :math:`\mathbf{q}`: distribution in the target space
- `L`: loss function to account for the misfit between the similarity matrices
- `H`: entropy
.. note:: If the inner solver `ot.sinkhorn` did not convergence, the
optimal coupling :math:`\mathbf{T}` returned by this function does not
necessarily satisfy the marginal constraints
:math:`\mathbf{T}\mathbf{1}=\mathbf{p}` and
:math:`\mathbf{T}^T\mathbf{1}=\mathbf{q}`. So the returned
Gromov-Wasserstein loss does not necessarily satisfy distance
properties and may be negative.
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
q : array-like, shape (nt,)
Distribution in the target space
loss_fun : str
Loss function used for the solver either 'square_loss' or 'kl_loss'
epsilon : float
Regularization term >0
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).
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 : float, optional
Stop threshold on error (>0)
verbose : bool, optional
Print information along iterations
log : bool, optional
Record log if True.
Returns
-------
gw_dist : float
Gromov-Wasserstein distance
References
----------
.. [12] Gabriel Peyré, Marco Cuturi, and Justin Solomon,
"Gromov-Wasserstein averaging of kernel and distance matrices."
International Conference on Machine Learning (ICML). 2016.
"""
gw, logv = entropic_gromov_wasserstein(
C1, C2, p, q, loss_fun, epsilon, symmetric, G0, max_iter, tol, verbose, log=True)
logv['T'] = gw
if log:
return logv['gw_dist'], logv
else:
return logv['gw_dist']
def entropic_gromov_barycenters(N, Cs, ps, p, lambdas, loss_fun, epsilon, symmetric=True,
max_iter=1000, tol=1e-9, verbose=False, log=False, init_C=None, random_state=None):
r"""
Returns the gromov-wasserstein barycenters of `S` measured similarity matrices :math:`(\mathbf{C}_s)_{1 \leq s \leq S}`
The function solves the following optimization problem:
.. math::
\mathbf{C} = \mathop{\arg \min}_{\mathbf{C}\in \mathbb{R}^{N \times N}} \quad \sum_s \lambda_s \mathrm{GW}(\mathbf{C}, \mathbf{C}_s, \mathbf{p}, \mathbf{p}_s)
Where :
- :math:`\mathbf{C}_s`: metric cost matrix
- :math:`\mathbf{p}_s`: distribution
Parameters
----------
N : int
Size of the targeted barycenter
Cs : list of S array-like of shape (ns,ns)
Metric cost matrices
ps : list of S array-like of shape (ns,)
Sample weights in the `S` spaces
p : array-like, shape(N,)
Weights in the targeted barycenter
lambdas : list of float
List of the `S` spaces' weights.
loss_fun : callable
Tensor-matrix multiplication function based on specific loss function.
update : callable
function(:math:`\mathbf{p}`, lambdas, :math:`\mathbf{T}`, :math:`\mathbf{Cs}`) that updates
:math:`\mathbf{C}` according to a specific Kernel with the `S` :math:`\mathbf{T}_s` couplings
calculated at each iteration
epsilon : float
Regularization term >0
symmetric : bool, optional.
Either structures are to be assumed symmetric or not. Default value is True.
Else if set to True (resp. False), C1 and C2 will be assumed symmetric (resp. asymmetric).
max_iter : int, optional
Max number of iterations
tol : float, optional
Stop threshold on error (>0)
verbose : bool, optional
Print information along iterations.
log : bool, optional
Record log if True.
init_C : bool | array-like, shape (N, N)
Random initial value for the :math:`\mathbf{C}` matrix provided by user.
random_state : int or RandomState instance, optional
Fix the seed for reproducibility
Returns
-------
C : array-like, shape (`N`, `N`)
Similarity matrix in the barycenter space (permutated arbitrarily)
log : dict
Log dictionary of error during iterations. Return only if `log=True` in parameters.
References
----------
.. [12] Gabriel Peyré, Marco Cuturi, and Justin Solomon,
"Gromov-Wasserstein averaging of kernel and distance matrices."
International Conference on Machine Learning (ICML). 2016.
"""
Cs = list_to_array(*Cs)
ps = list_to_array(*ps)
p = list_to_array(p)
nx = get_backend(*Cs, *ps, p)
S = len(Cs)
# Initialization of C : random SPD matrix (if not provided by user)
if init_C is None:
generator = check_random_state(random_state)
xalea = generator.randn(N, 2)
C = dist(xalea, xalea)
C /= C.max()
C = nx.from_numpy(C, type_as=p)
else:
C = init_C
cpt = 0
err = 1
error = []
while (err > tol) and (cpt < max_iter):
Cprev = C
T = [entropic_gromov_wasserstein(Cs[s], C, ps[s], p, loss_fun, epsilon, symmetric, None,
max_iter, 1e-4, verbose, log=False) for s in range(S)]
if loss_fun == 'square_loss':
C = update_square_loss(p, lambdas, T, Cs)
elif loss_fun == 'kl_loss':
C = update_kl_loss(p, lambdas, T, Cs)
if cpt % 10 == 0:
# we can speed up the process by checking for the error only all
# the 10th iterations
err = nx.norm(C - Cprev)
error.append(err)
if verbose:
if cpt % 200 == 0:
print('{:5s}|{:12s}'.format(
'It.', 'Err') + '\n' + '-' * 19)
print('{:5d}|{:8e}|'.format(cpt, err))
cpt += 1
if log:
return C, {"err": error}
else:
return C
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