https://github.com/GPflow/GPflow
Tip revision: a806b62d30187dc6c831b4eb569d98c0c19ad60c authored by Artem Artemev on 01 January 2018, 16:52:19 UTC
Initiate porting from testing environment.
Initiate porting from testing environment.
Tip revision: a806b62
kullback_leiblers.py
# Copyright 2016 James Hensman, alexggmatthews
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
# -*- coding: utf-8 -*-
import tensorflow as tf
from . import settings
from .decors import name_scope
@name_scope()
def gauss_kl(q_mu, q_sqrt, K=None):
"""
Compute the KL divergence KL[q || p] between
q(x) = N(q_mu, q_sqrt^2)
and
p(x) = N(0, K)
We assume N multiple independent distributions, given by the columns of
q_mu and the last dimension of q_sqrt. Returns the sum of the divergences.
q_mu is a matrix (M x N), each column contains a mean.
q_sqrt can be a 3D tensor (M x M x N), each matrix within is a lower
triangular square-root matrix of the covariance of q.
q_sqrt can be a matrix (M x N), each column represents the diagonal of a
square-root matrix of the covariance of q.
K is a positive definite matrix (M x M): the covariance of p.
If K is None, compute the KL divergence to p(x) = N(0, I) instead.
"""
if K is None:
white = True
alpha = q_mu
else:
white = False
Lp = tf.cholesky(K)
alpha = tf.matrix_triangular_solve(Lp, q_mu, lower=True)
if q_sqrt.get_shape().ndims == 2:
diag = True
num_latent = tf.shape(q_sqrt)[1]
NM = tf.size(q_sqrt)
Lq = Lq_diag = q_sqrt
elif q_sqrt.get_shape().ndims == 3:
diag = False
num_latent = tf.shape(q_sqrt)[2]
NM = tf.reduce_prod(tf.shape(q_sqrt)[1:])
Lq = tf.matrix_band_part(tf.transpose(q_sqrt, (2, 0, 1)), -1, 0) # force lower triangle
Lq_diag = tf.matrix_diag_part(Lq)
else: # pragma: no cover
raise ValueError("Bad dimension for q_sqrt: {}".format(q_sqrt.get_shape().ndims))
# Mahalanobis term: μqᵀ Σp⁻¹ μq
mahalanobis = tf.reduce_sum(tf.square(alpha))
# Constant term: - N x M
constant = - tf.cast(NM, settings.float_type)
# Log-determinant of the covariance of q(x):
logdet_qcov = tf.reduce_sum(tf.log(tf.square(Lq_diag)))
# Trace term: tr(Σp⁻¹ Σq)
if white:
trace = tf.reduce_sum(tf.square(Lq))
else:
if diag:
M = tf.shape(Lp)[0]
Lp_inv = tf.matrix_triangular_solve(
Lp, tf.eye(M, dtype=settings.float_type), lower=True)
K_inv = tf.matrix_triangular_solve(
tf.transpose(Lp), Lp_inv, lower=False)
trace = tf.reduce_sum(
tf.expand_dims(tf.matrix_diag_part(K_inv), 1) * tf.square(q_sqrt))
else:
Lp_tiled = tf.tile(tf.expand_dims(Lp, 0), [num_latent, 1, 1])
LpiLq = tf.matrix_triangular_solve(Lp_tiled, Lq, lower=True)
trace = tf.reduce_sum(tf.square(LpiLq))
twoKL = mahalanobis + constant - logdet_qcov + trace
# Log-determinant of the covariance of p(x):
if not white:
log_sqdiag_Lp = tf.log(tf.square(tf.matrix_diag_part(Lp)))
sum_log_sqdiag_Lp = tf.reduce_sum(log_sqdiag_Lp)
prior_logdet = tf.cast(num_latent, settings.float_type) * sum_log_sqdiag_Lp
twoKL += prior_logdet
return 0.5 * twoKL