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# -*- coding: utf-8 -*-
"""
Optimal transport for Gaussian distributions
"""
# Author: Theo Gnassounou <theo.gnassounou@inria.fr>
# Remi Flamary <remi.flamary@polytehnique.edu>
#
# License: MIT License
from .backend import get_backend
from .utils import dots
from .utils import list_to_array
def bures_wasserstein_mapping(ms, mt, Cs, Ct, log=False):
r"""Return OT linear operator between samples.
The function estimates the optimal linear operator that aligns the two
empirical distributions. This is equivalent to estimating the closed
form mapping between two Gaussian distributions :math:`\mathcal{N}(\mu_s,\Sigma_s)`
and :math:`\mathcal{N}(\mu_t,\Sigma_t)` as proposed in
:ref:`[1] <references-OT-mapping-linear>` and discussed in remark 2.29 in
:ref:`[2] <references-OT-mapping-linear>`.
The linear operator from source to target :math:`M`
.. math::
M(\mathbf{x})= \mathbf{A} \mathbf{x} + \mathbf{b}
where :
.. math::
\mathbf{A} &= \Sigma_s^{-1/2} \left(\Sigma_s^{1/2}\Sigma_t\Sigma_s^{1/2} \right)^{1/2}
\Sigma_s^{-1/2}
\mathbf{b} &= \mu_t - \mathbf{A} \mu_s
Parameters
----------
ms : array-like (d,)
mean of the source distribution
mt : array-like (d,)
mean of the target distribution
Cs : array-like (d,)
covariance of the source distribution
Ct : array-like (d,)
covariance of the target distribution
log : bool, optional
record log if True
Returns
-------
A : (d, d) array-like
Linear operator
b : (1, d) array-like
bias
log : dict
log dictionary return only if log==True in parameters
.. _references-OT-mapping-linear:
References
----------
.. [1] Knott, M. and Smith, C. S. "On the optimal mapping of
distributions", Journal of Optimization Theory and Applications
Vol 43, 1984
.. [2] Peyré, G., & Cuturi, M. (2017). "Computational Optimal
Transport", 2018.
"""
ms, mt, Cs, Ct = list_to_array(ms, mt, Cs, Ct)
nx = get_backend(ms, mt, Cs, Ct)
Cs12 = nx.sqrtm(Cs)
Cs12inv = nx.inv(Cs12)
M0 = nx.sqrtm(dots(Cs12, Ct, Cs12))
A = dots(Cs12inv, M0, Cs12inv)
b = mt - nx.dot(ms, A)
if log:
log = {}
log['Cs12'] = Cs12
log['Cs12inv'] = Cs12inv
return A, b, log
else:
return A, b
def empirical_bures_wasserstein_mapping(xs, xt, reg=1e-6, ws=None,
wt=None, bias=True, log=False):
r"""Return OT linear operator between samples.
The function estimates the optimal linear operator that aligns the two
empirical distributions. This is equivalent to estimating the closed
form mapping between two Gaussian distributions :math:`\mathcal{N}(\mu_s,\Sigma_s)`
and :math:`\mathcal{N}(\mu_t,\Sigma_t)` as proposed in
:ref:`[1] <references-OT-mapping-linear>` and discussed in remark 2.29 in
:ref:`[2] <references-OT-mapping-linear>`.
The linear operator from source to target :math:`M`
.. math::
M(\mathbf{x})= \mathbf{A} \mathbf{x} + \mathbf{b}
where :
.. math::
\mathbf{A} &= \Sigma_s^{-1/2} \left(\Sigma_s^{1/2}\Sigma_t\Sigma_s^{1/2} \right)^{1/2}
\Sigma_s^{-1/2}
\mathbf{b} &= \mu_t - \mathbf{A} \mu_s
Parameters
----------
xs : array-like (ns,d)
samples in the source domain
xt : array-like (nt,d)
samples in the target domain
reg : float,optional
regularization added to the diagonals of covariances (>0)
ws : array-like (ns,1), optional
weights for the source samples
wt : array-like (ns,1), optional
weights for the target samples
bias: boolean, optional
estimate bias :math:`\mathbf{b}` else :math:`\mathbf{b} = 0` (default:True)
log : bool, optional
record log if True
Returns
-------
A : (d, d) array-like
Linear operator
b : (1, d) array-like
bias
log : dict
log dictionary return only if log==True in parameters
.. _references-OT-mapping-linear:
References
----------
.. [1] Knott, M. and Smith, C. S. "On the optimal mapping of
distributions", Journal of Optimization Theory and Applications
Vol 43, 1984
.. [2] Peyré, G., & Cuturi, M. (2017). "Computational Optimal
Transport", 2018.
"""
xs, xt = list_to_array(xs, xt)
nx = get_backend(xs, xt)
d = xs.shape[1]
if bias:
mxs = nx.mean(xs, axis=0)[None, :]
mxt = nx.mean(xt, axis=0)[None, :]
xs = xs - mxs
xt = xt - mxt
else:
mxs = nx.zeros((1, d), type_as=xs)
mxt = nx.zeros((1, d), type_as=xs)
if ws is None:
ws = nx.ones((xs.shape[0], 1), type_as=xs) / xs.shape[0]
if wt is None:
wt = nx.ones((xt.shape[0], 1), type_as=xt) / xt.shape[0]
Cs = nx.dot((xs * ws).T, xs) / nx.sum(ws) + reg * nx.eye(d, type_as=xs)
Ct = nx.dot((xt * wt).T, xt) / nx.sum(wt) + reg * nx.eye(d, type_as=xt)
if log:
A, b, log = bures_wasserstein_mapping(mxs, mxt, Cs, Ct, log=log)
log['Cs'] = Cs
log['Ct'] = Ct
return A, b, log
else:
A, b = bures_wasserstein_mapping(mxs, mxt, Cs, Ct)
return A, b
def bures_wasserstein_distance(ms, mt, Cs, Ct, log=False):
r"""Return Bures Wasserstein distance between samples.
The function estimates the Bures-Wasserstein distance between two
empirical distributions source :math:`\mu_s` and target :math:`\mu_t`,
discussed in remark 2.31 :ref:`[1] <references-bures-wasserstein-distance>`.
The Bures Wasserstein distance between source and target distribution :math:`\mathcal{W}`
.. math::
\mathcal{W}(\mu_s, \mu_t)_2^2= \left\lVert \mathbf{m}_s - \mathbf{m}_t \right\rVert^2 + \mathcal{B}(\Sigma_s, \Sigma_t)^{2}
where :
.. math::
\mathbf{B}(\Sigma_s, \Sigma_t)^{2} = \text{Tr}\left(\Sigma_s^{1/2} + \Sigma_t^{1/2} - 2 \sqrt{\Sigma_s^{1/2}\Sigma_t\Sigma_s^{1/2}} \right)
Parameters
----------
ms : array-like (d,)
mean of the source distribution
mt : array-like (d,)
mean of the target distribution
Cs : array-like (d,)
covariance of the source distribution
Ct : array-like (d,)
covariance of the target distribution
log : bool, optional
record log if True
Returns
-------
W : float
Bures Wasserstein distance
log : dict
log dictionary return only if log==True in parameters
.. _references-bures-wasserstein-distance:
References
----------
.. [1] Peyré, G., & Cuturi, M. (2017). "Computational Optimal
Transport", 2018.
"""
ms, mt, Cs, Ct = list_to_array(ms, mt, Cs, Ct)
nx = get_backend(ms, mt, Cs, Ct)
Cs12 = nx.sqrtm(Cs)
B = nx.trace(Cs + Ct - 2 * nx.sqrtm(dots(Cs12, Ct, Cs12)))
W = nx.sqrt(nx.norm(ms - mt)**2 + B)
if log:
log = {}
log['Cs12'] = Cs12
return W, log
else:
return W
def empirical_bures_wasserstein_distance(xs, xt, reg=1e-6, ws=None,
wt=None, bias=True, log=False):
r"""Return Bures Wasserstein distance from mean and covariance of distribution.
The function estimates the Bures-Wasserstein distance between two
empirical distributions source :math:`\mu_s` and target :math:`\mu_t`,
discussed in remark 2.31 :ref:`[1] <references-bures-wasserstein-distance>`.
The Bures Wasserstein distance between source and target distribution :math:`\mathcal{W}`
.. math::
\mathcal{W}(\mu_s, \mu_t)_2^2= \left\lVert \mathbf{m}_s - \mathbf{m}_t \right\rVert^2 + \mathcal{B}(\Sigma_s, \Sigma_t)^{2}
where :
.. math::
\mathbf{B}(\Sigma_s, \Sigma_t)^{2} = \text{Tr}\left(\Sigma_s^{1/2} + \Sigma_t^{1/2} - 2 \sqrt{\Sigma_s^{1/2}\Sigma_t\Sigma_s^{1/2}} \right)
Parameters
----------
xs : array-like (ns,d)
samples in the source domain
xt : array-like (nt,d)
samples in the target domain
reg : float,optional
regularization added to the diagonals of covariances (>0)
ws : array-like (ns,1), optional
weights for the source samples
wt : array-like (ns,1), optional
weights for the target samples
bias: boolean, optional
estimate bias :math:`\mathbf{b}` else :math:`\mathbf{b} = 0` (default:True)
log : bool, optional
record log if True
Returns
-------
W : float
Bures Wasserstein distance
log : dict
log dictionary return only if log==True in parameters
.. _references-bures-wasserstein-distance:
References
----------
.. [1] Peyré, G., & Cuturi, M. (2017). "Computational Optimal
Transport", 2018.
"""
xs, xt = list_to_array(xs, xt)
nx = get_backend(xs, xt)
d = xs.shape[1]
if bias:
mxs = nx.mean(xs, axis=0)[None, :]
mxt = nx.mean(xt, axis=0)[None, :]
xs = xs - mxs
xt = xt - mxt
else:
mxs = nx.zeros((1, d), type_as=xs)
mxt = nx.zeros((1, d), type_as=xs)
if ws is None:
ws = nx.ones((xs.shape[0], 1), type_as=xs) / xs.shape[0]
if wt is None:
wt = nx.ones((xt.shape[0], 1), type_as=xt) / xt.shape[0]
Cs = nx.dot((xs * ws).T, xs) / nx.sum(ws) + reg * nx.eye(d, type_as=xs)
Ct = nx.dot((xt * wt).T, xt) / nx.sum(wt) + reg * nx.eye(d, type_as=xt)
if log:
W, log = bures_wasserstein_distance(mxs, mxt, Cs, Ct, log=log)
log['Cs'] = Cs
log['Ct'] = Ct
return W, log
else:
W = bures_wasserstein_distance(mxs, mxt, Cs, Ct)
return W
|
