Revision

**291ae6c7dbfcbded27c604f136982a5067d14b8e**authored by thevincentadam on**20 January 2020, 12:17:20 UTC**, committed by thevincentadam on**20 January 2020, 12:17:20 UTC****1 parent**5dc31b8

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 .config import default_float
from multipledispatch import Dispatcher
from .inducing_variables import InducingVariables
from .kernels import Kernel
from .covariances.kuus import Kuu
from .config import default_jitter
prior_kl = Dispatcher('prior_kl')
@prior_kl.register(InducingVariables, Kernel, object, object)
def _(inducing_variable, kernel, q_mu, q_sqrt, whiten=False):
if whiten:
return gauss_kl(q_mu, q_sqrt, None)
else:
K = Kuu(inducing_variable, kernel, jitter=default_jitter()) # [P, M, M] or [M, M]
return gauss_kl(q_mu, q_sqrt, K)
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) if K is not None
p(x) = N(0, I) if K is None
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, L]), each column contains a mean.
q_sqrt can be a 3D tensor ([L, M, M]), each matrix within is a lower
triangular square-root matrix of the covariance of q.
q_sqrt can be a matrix ([M, L]), each column represents the diagonal of a
square-root matrix of the covariance of q.
K is the covariance of p.
It is a positive definite matrix ([M, M]) or a tensor of stacked such matrices ([L, M, M])
If K is None, compute the KL divergence to p(x) = N(0, I) instead.
"""
white = K is None
diag = len(q_sqrt.shape) == 2
M, B = tf.shape(q_mu)[0], tf.shape(q_mu)[1]
if white:
alpha = q_mu # [M, B]
else:
batch = len(K.shape) == 3
Lp = tf.linalg.cholesky(K) # [B, M, M] or [M, M]
q_mu = tf.transpose(
q_mu)[:, :, None] if batch else q_mu # [B, M, 1] or [M, B]
alpha = tf.linalg.triangular_solve(Lp, q_mu,
lower=True) # [B, M, 1] or [M, B]
if diag:
Lq = Lq_diag = q_sqrt
Lq_full = tf.linalg.diag(tf.transpose(q_sqrt)) # [B, M, M]
else:
Lq = Lq_full = tf.linalg.band_part(
q_sqrt, -1, 0) # force lower triangle # [B, M, M]
Lq_diag = tf.linalg.diag_part(Lq) # [M, B]
# Mahalanobis term: μqᵀ Σp⁻¹ μq
mahalanobis = tf.reduce_sum(tf.square(alpha))
# Constant term: - B * M
constant = -tf.cast(tf.size(q_mu, out_type=tf.int64),
dtype=default_float())
# Log-determinant of the covariance of q(x):
logdet_qcov = tf.reduce_sum(tf.math.log(tf.square(Lq_diag)))
# Trace term: tr(Σp⁻¹ Σq)
if white:
trace = tf.reduce_sum(tf.square(Lq))
else:
if diag and not batch:
# K is [M, M] and q_sqrt is [M, B]: fast specialisation
LpT = tf.transpose(Lp) # [M, M]
Lp_inv = tf.linalg.triangular_solve(Lp,
tf.eye(M,
dtype=default_float()),
lower=True) # [M, M]
K_inv = tf.linalg.diag_part(
tf.linalg.triangular_solve(
LpT, Lp_inv, lower=False))[:, None] # [M, M] -> [M, 1]
trace = tf.reduce_sum(K_inv * tf.square(q_sqrt))
else:
# TODO: broadcast instead of tile when tf allows (not implemented in tf <= 1.6.0)
Lp_full = Lp if batch else tf.tile(tf.expand_dims(Lp, 0),
[B, 1, 1])
LpiLq = tf.linalg.triangular_solve(Lp_full, Lq_full, 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.math.log(tf.square(tf.linalg.diag_part(Lp)))
sum_log_sqdiag_Lp = tf.reduce_sum(log_sqdiag_Lp)
# If K is [B, M, M], num_latent is no longer implicit, no need to multiply the single kernel logdet
scale = 1.0 if batch else tf.cast(B, default_float())
twoKL += scale * sum_log_sqdiag_Lp
return 0.5 * twoKL
```

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