# Copyright 2016 James Hensman, alexggmatthews, Mark van der Wilk
#
# 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.
from __future__ import absolute_import
import tensorflow as tf
import numpy as np
from .model import GPModel
from .param import Param, DataHolder, AutoFlow
from .mean_functions import Zero
from . import likelihoods
from ._settings import settings
float_type = settings.dtypes.float_type
class SGPRUpperMixin(object):
"""
Upper bound for the GP regression marginal likelihood.
It is implemented here as a Mixin class which works with SGPR and GPRFITC. Note that the same inducing points are
used for calculating the upper bound, as are used for computing the likelihood approximation. This may not lead to
the best upper bound. The upper bound can be tightened by optimising Z, just as just like the lower bound. This is
especially important in FITC, as FITC is known to produce poor inducing point locations. An optimisable upper bound
can be found in https://github.com/markvdw/gp_upper.
The key reference is
::
@misc{titsias_2014,
title={Variational Inference for Gaussian and Determinantal Point Processes},
url={http://www2.aueb.gr/users/mtitsias/papers/titsiasNipsVar14.pdf},
publisher={Workshop on Advances in Variational Inference (NIPS 2014)},
author={Titsias, Michalis K.},
year={2014},
month={Dec}
}
"""
@AutoFlow()
def compute_upper_bound(self):
num_inducing = tf.shape(self.Z)[0]
num_data = tf.cast(tf.shape(self.Y)[0], float_type)
Kdiag = self.kern.Kdiag(self.X)
Kuu = self.kern.K(self.Z) + tf.eye(num_inducing, dtype=float_type) * settings.numerics.jitter_level
Kuf = self.kern.K(self.Z, self.X)
L = tf.cholesky(Kuu)
LB = tf.cholesky(Kuu + self.likelihood.variance ** -1.0 * tf.matmul(Kuf, Kuf, transpose_b=True))
LinvKuf = tf.matrix_triangular_solve(L, Kuf, lower=True)
c = tf.reduce_sum(Kdiag) - tf.reduce_sum(LinvKuf ** 2.0) # Using the Trace bound, from Titsias' presentation
# Kff = self.kern.K(self.X)
# Qff = tf.matmul(Kuf, LinvKuf, transpose_a=True)
# c = tf.reduce_max(tf.reduce_sum(tf.abs(Kff - Qff), 0)) # Alternative bound on max eigenval
corrected_noise = self.likelihood.variance + c
const = -0.5 * num_data * tf.log(2 * np.pi * self.likelihood.variance)
logdet = tf.reduce_sum(tf.log(tf.diag_part(L))) - tf.reduce_sum(tf.log(tf.diag_part(LB)))
LC = tf.cholesky(Kuu + corrected_noise ** -1.0 * tf.matmul(Kuf, Kuf, transpose_b=True))
v = tf.matrix_triangular_solve(LC, corrected_noise ** -1.0 * tf.matmul(Kuf, self.Y), lower=True)
quad = -0.5 * corrected_noise ** -1.0 * tf.reduce_sum(self.Y ** 2.0) + 0.5 * tf.reduce_sum(v ** 2.0)
return const + logdet + quad
class SGPR(GPModel, SGPRUpperMixin):
"""
Sparse Variational GP regression. The key reference is
::
@inproceedings{titsias2009variational,
title={Variational learning of inducing variables in
sparse Gaussian processes},
author={Titsias, Michalis K},
booktitle={International Conference on
Artificial Intelligence and Statistics},
pages={567--574},
year={2009}
}
"""
def __init__(self, X, Y, kern, Z, mean_function=None):
"""
X is a data matrix, size N x D
Y is a data matrix, size N x R
Z is a matrix of pseudo inputs, size M x D
kern, mean_function are appropriate GPflow objects
This method only works with a Gaussian likelihood.
"""
X = DataHolder(X, on_shape_change='pass')
Y = DataHolder(Y, on_shape_change='pass')
likelihood = likelihoods.Gaussian()
GPModel.__init__(self, X, Y, kern, likelihood, mean_function)
self.Z = Param(Z)
self.num_data = X.shape[0]
self.num_latent = Y.shape[1]
def build_likelihood(self):
"""
Construct a tensorflow function to compute the bound on the marginal
likelihood. For a derivation of the terms in here, see the associated
SGPR notebook.
"""
num_inducing = tf.shape(self.Z)[0]
num_data = tf.cast(tf.shape(self.Y)[0], settings.dtypes.float_type)
output_dim = tf.cast(tf.shape(self.Y)[1], settings.dtypes.float_type)
err = self.Y - self.mean_function(self.X)
Kdiag = self.kern.Kdiag(self.X)
Kuf = self.kern.K(self.Z, self.X)
Kuu = self.kern.K(self.Z) + tf.eye(num_inducing, dtype=float_type) * settings.numerics.jitter_level
L = tf.cholesky(Kuu)
sigma = tf.sqrt(self.likelihood.variance)
# Compute intermediate matrices
A = tf.matrix_triangular_solve(L, Kuf, lower=True) / sigma
AAT = tf.matmul(A, A, transpose_b=True)
B = AAT + tf.eye(num_inducing, dtype=float_type)
LB = tf.cholesky(B)
Aerr = tf.matmul(A, err)
c = tf.matrix_triangular_solve(LB, Aerr, lower=True) / sigma
# compute log marginal bound
bound = -0.5 * num_data * output_dim * np.log(2 * np.pi)
bound += - output_dim * tf.reduce_sum(tf.log(tf.matrix_diag_part(LB)))
bound -= 0.5 * num_data * output_dim * tf.log(self.likelihood.variance)
bound += -0.5 * tf.reduce_sum(tf.square(err)) / self.likelihood.variance
bound += 0.5 * tf.reduce_sum(tf.square(c))
bound += -0.5 * output_dim * tf.reduce_sum(Kdiag) / self.likelihood.variance
bound += 0.5 * output_dim * tf.reduce_sum(tf.matrix_diag_part(AAT))
return bound
def build_predict(self, Xnew, full_cov=False):
"""
Compute the mean and variance of the latent function at some new points
Xnew. For a derivation of the terms in here, see the associated SGPR
notebook.
"""
num_inducing = tf.shape(self.Z)[0]
err = self.Y - self.mean_function(self.X)
Kuf = self.kern.K(self.Z, self.X)
Kuu = self.kern.K(self.Z) + tf.eye(num_inducing, dtype=float_type) * settings.numerics.jitter_level
Kus = self.kern.K(self.Z, Xnew)
sigma = tf.sqrt(self.likelihood.variance)
L = tf.cholesky(Kuu)
A = tf.matrix_triangular_solve(L, Kuf, lower=True) / sigma
B = tf.matmul(A, A, transpose_b=True) + tf.eye(num_inducing, dtype=float_type)
LB = tf.cholesky(B)
Aerr = tf.matmul(A, err)
c = tf.matrix_triangular_solve(LB, Aerr, lower=True) / sigma
tmp1 = tf.matrix_triangular_solve(L, Kus, lower=True)
tmp2 = tf.matrix_triangular_solve(LB, tmp1, lower=True)
mean = tf.matmul(tmp2, c, transpose_a=True)
if full_cov:
var = self.kern.K(Xnew) + tf.matmul(tmp2, tmp2, transpose_a=True) \
- tf.matmul(tmp1, tmp1, transpose_a=True)
shape = tf.stack([1, 1, tf.shape(self.Y)[1]])
var = tf.tile(tf.expand_dims(var, 2), shape)
else:
var = self.kern.Kdiag(Xnew) + tf.reduce_sum(tf.square(tmp2), 0) \
- tf.reduce_sum(tf.square(tmp1), 0)
shape = tf.stack([1, tf.shape(self.Y)[1]])
var = tf.tile(tf.expand_dims(var, 1), shape)
return mean + self.mean_function(Xnew), var
class GPRFITC(GPModel, SGPRUpperMixin):
def __init__(self, X, Y, kern, Z, mean_function=Zero()):
"""
This implements GP regression with the FITC approximation.
The key reference is
@inproceedings{Snelson06sparsegaussian,
author = {Edward Snelson and Zoubin Ghahramani},
title = {Sparse Gaussian Processes using Pseudo-inputs},
booktitle = {Advances In Neural Information Processing Systems },
year = {2006},
pages = {1257--1264},
publisher = {MIT press}
}
Implementation loosely based on code from GPML matlab library although
obviously gradients are automatic in GPflow.
X is a data matrix, size N x D
Y is a data matrix, size N x R
Z is a matrix of pseudo inputs, size M x D
kern, mean_function are appropriate GPflow objects
This method only works with a Gaussian likelihood.
"""
X = DataHolder(X, on_shape_change='pass')
Y = DataHolder(Y, on_shape_change='pass')
likelihood = likelihoods.Gaussian()
GPModel.__init__(self, X, Y, kern, likelihood, mean_function)
self.Z = Param(Z)
self.num_data = X.shape[0]
self.num_latent = Y.shape[1]
def build_common_terms(self):
num_inducing = tf.shape(self.Z)[0]
err = self.Y - self.mean_function(self.X) # size N x R
Kdiag = self.kern.Kdiag(self.X)
Kuf = self.kern.K(self.Z, self.X)
Kuu = self.kern.K(self.Z) + tf.eye(num_inducing, dtype=float_type) * settings.numerics.jitter_level
Luu = tf.cholesky(Kuu) # => Luu Luu^T = Kuu
V = tf.matrix_triangular_solve(Luu, Kuf) # => V^T V = Qff = Kuf^T Kuu^-1 Kuf
diagQff = tf.reduce_sum(tf.square(V), 0)
nu = Kdiag - diagQff + self.likelihood.variance
B = tf.eye(num_inducing, dtype=float_type) + tf.matmul(V / nu, V, transpose_b=True)
L = tf.cholesky(B)
beta = err / tf.expand_dims(nu, 1) # size N x R
alpha = tf.matmul(V, beta) # size N x R
gamma = tf.matrix_triangular_solve(L, alpha, lower=True) # size N x R
return err, nu, Luu, L, alpha, beta, gamma
def build_likelihood(self):
"""
Construct a tensorflow function to compute the bound on the marginal
likelihood.
"""
# FITC approximation to the log marginal likelihood is
# log ( normal( y | mean, K_fitc ) )
# where K_fitc = Qff + diag( \nu )
# where Qff = Kfu Kuu^{-1} Kuf
# with \nu_i = Kff_{i,i} - Qff_{i,i} + \sigma^2
# We need to compute the Mahalanobis term -0.5* err^T K_fitc^{-1} err
# (summed over functions).
# We need to deal with the matrix inverse term.
# K_fitc^{-1} = ( Qff + \diag( \nu ) )^{-1}
# = ( V^T V + \diag( \nu ) )^{-1}
# Applying the Woodbury identity we obtain
# = \diag( \nu^{-1} ) - \diag( \nu^{-1} ) V^T ( I + V \diag( \nu^{-1} ) V^T )^{-1) V \diag(\nu^{-1} )
# Let \beta = \diag( \nu^{-1} ) err
# and let \alpha = V \beta
# then Mahalanobis term = -0.5* ( \beta^T err - \alpha^T Solve( I + V \diag( \nu^{-1} ) V^T, alpha ) )
err, nu, Luu, L, alpha, beta, gamma = self.build_common_terms()
mahalanobisTerm = -0.5 * tf.reduce_sum(tf.square(err) / tf.expand_dims(nu, 1)) \
+ 0.5 * tf.reduce_sum(tf.square(gamma))
# We need to compute the log normalizing term -N/2 \log 2 pi - 0.5 \log \det( K_fitc )
# We need to deal with the log determinant term.
# \log \det( K_fitc ) = \log \det( Qff + \diag( \nu ) )
# = \log \det( V^T V + \diag( \nu ) )
# Applying the determinant lemma we obtain
# = \log [ \det \diag( \nu ) \det( I + V \diag( \nu^{-1} ) V^T ) ]
# = \log [ \det \diag( \nu ) ] + \log [ \det( I + V \diag( \nu^{-1} ) V^T ) ]
constantTerm = -0.5 * self.num_data * tf.log(tf.constant(2. * np.pi, settings.dtypes.float_type))
logDeterminantTerm = -0.5 * tf.reduce_sum(tf.log(nu)) - tf.reduce_sum(tf.log(tf.matrix_diag_part(L)))
logNormalizingTerm = constantTerm + logDeterminantTerm
return mahalanobisTerm + logNormalizingTerm * self.num_latent
def build_predict(self, Xnew, full_cov=False):
"""
Compute the mean and variance of the latent function at some new points
Xnew.
"""
_, _, Luu, L, _, _, gamma = self.build_common_terms()
Kus = self.kern.K(self.Z, Xnew) # size M x Xnew
w = tf.matrix_triangular_solve(Luu, Kus, lower=True) # size M x Xnew
tmp = tf.matrix_triangular_solve(tf.transpose(L), gamma, lower=False)
mean = tf.matmul(w, tmp, transpose_a=True) + self.mean_function(Xnew)
intermediateA = tf.matrix_triangular_solve(L, w, lower=True)
if full_cov:
var = self.kern.K(Xnew) - tf.matmul(w, w, transpose_a=True) \
+ tf.matmul(intermediateA, intermediateA, transpose_a=True)
var = tf.tile(tf.expand_dims(var, 2), tf.stack([1, 1, tf.shape(self.Y)[1]]))
else:
var = self.kern.Kdiag(Xnew) - tf.reduce_sum(tf.square(w), 0) \
+ tf.reduce_sum(tf.square(intermediateA), 0) # size Xnew,
var = tf.tile(tf.expand_dims(var, 1), tf.stack([1, tf.shape(self.Y)[1]]))
return mean, var
