"""Distance utils."""
import numpy
from scipy.linalg import eigvalsh
from .base import logm, sqrtm
def distance_kullback(A, B):
"""Kullback leibler divergence between two covariance matrices A and B.
:param A: First covariance matrix
:param B: Second covariance matrix
:returns: Kullback leibler divergence between A and B
"""
dim = A.shape[0]
logdet = numpy.log(numpy.linalg.det(B) / numpy.linalg.det(A))
kl = numpy.trace(numpy.dot(numpy.linalg.inv(B), A)) - dim + logdet
return 0.5 * kl
def distance_kullback_right(A, B):
"""wrapper for right kullblack leibler div."""
return distance_kullback(B, A)
def distance_kullback_sym(A, B):
"""Symetrized kullback leibler divergence."""
return distance_kullback(A, B) + distance_kullback_right(A, B)
def distance_euclid(A, B):
"""Euclidean distance between two covariance matrices A and B.
The Euclidean distance is defined by the Froebenius norm between the two
matrices.
.. math::
d = \Vert \mathbf{A} - \mathbf{B} \Vert_F
:param A: First covariance matrix
:param B: Second covariance matrix
:returns: Eclidean distance between A and B
"""
return numpy.linalg.norm(A - B, ord='fro')
def distance_logeuclid(A, B):
"""Log Euclidean distance between two covariance matrices A and B.
.. math::
d = \Vert \log(\mathbf{A}) - \log(\mathbf{B}) \Vert_F
:param A: First covariance matrix
:param B: Second covariance matrix
:returns: Log-Eclidean distance between A and B
"""
return distance_euclid(logm(A), logm(B))
def distance_riemann(A, B):
"""Riemannian distance between two covariance matrices A and B.
.. math::
d = {\left( \sum_i \log(\lambda_i)^2 \\right)}^{-1/2}
where :math:`\lambda_i` are the joint eigenvalues of A and B
:param A: First covariance matrix
:param B: Second covariance matrix
:returns: Riemannian distance between A and B
"""
return numpy.sqrt((numpy.log(eigvalsh(A, B))**2).sum())
def distance_logdet(A, B):
"""Log-det distance between two covariance matrices A and B.
.. math::
d = \sqrt{\left(\log(\det(\\frac{\mathbf{A}+\mathbf{B}}{2})) - 0.5 \\times \log(\det(\mathbf{A}) \det(\mathbf{B}))\\right)}
:param A: First covariance matrix
:param B: Second covariance matrix
:returns: Log-Euclid distance between A and B
"""
return numpy.sqrt(numpy.log(numpy.linalg.det(
(A + B) / 2.0)) - 0.5 * numpy.log(numpy.linalg.det(A)*numpy.linalg.det(B)))
def distance_wasserstein(A, B):
"""Wasserstein distance between two covariances matrices.
.. math::
d = \left( {tr(A + B - 2(A^{1/2}BA^{1/2})^{1/2})}\\right )^{1/2}
:param A: First covariance matrix
:param B: Second covariance matrix
:returns: Wasserstein distance between A and B
"""
B12 = sqrtm(B)
C = sqrtm(numpy.dot(numpy.dot(B12, A), B12))
return numpy.sqrt(numpy.trace(A + B - 2*C))
def distance(A, B, metric='riemann'):
"""Distance between two covariance matrices A and B according to the metric.
:param A: First covariance matrix
:param B: Second covariance matrix
:param metric: the metric (Default value 'riemann'), can be : 'riemann' ,
'logeuclid' , 'euclid' , 'logdet', 'kullback', 'kullback_right',
'kullback_sym'.
:returns: the distance between A and B
"""
distance_methods = {'riemann': distance_riemann,
'logeuclid': distance_logeuclid,
'euclid': distance_euclid,
'logdet': distance_logdet,
'kullback': distance_kullback,
'kullback_right': distance_kullback_right,
'kullback_sym': distance_kullback_sym,
'wasserstein': distance_wasserstein}
if len(A.shape) == 3:
d = numpy.empty((len(A), 1))
for i in range(len(A)):
d[i] = distance_methods[metric](A[i], B)
else:
d = distance_methods[metric](A, B)
return d
