# 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 typing import Optional, Tuple
import numpy as np
import tensorflow as tf
from gpflow.kernels import Kernel
from .model import GPModel, InputData, RegressionData, MeanAndVariance
from .training_mixins import InternalDataTrainingLossMixin
from .util import inducingpoint_wrapper
from .. import likelihoods
from ..base import Parameter
from ..config import default_float, default_jitter
from ..covariances.dispatch import Kuf, Kuu
from ..inducing_variables import InducingPoints
from ..mean_functions import MeanFunction
from ..utilities import to_default_float, positive
class SGPRBase(GPModel, InternalDataTrainingLossMixin):
"""
Common base class for SGPR and GPRFITC that provides the common __init__
and upper_bound() methods.
"""
def __init__(
self,
data: RegressionData,
kernel: Kernel,
inducing_variable: InducingPoints,
*,
mean_function: Optional[MeanFunction] = None,
num_latent_gps: Optional[int] = None,
noise_variance: float = 1.0,
):
"""
`data`: a tuple of (X, Y), where the inputs X has shape [N, D]
and the outputs Y has shape [N, R].
`inducing_variable`: an InducingPoints instance or a matrix of
the pseudo inputs Z, of shape [M, D].
`kernel`, `mean_function` are appropriate GPflow objects
This method only works with a Gaussian likelihood, its variance is
initialized to `noise_variance`.
"""
likelihood = likelihoods.Gaussian(noise_variance)
X_data, Y_data = data
num_latent_gps = Y_data.shape[-1] if num_latent_gps is None else num_latent_gps
super().__init__(kernel, likelihood, mean_function, num_latent_gps=num_latent_gps)
self.data = data
self.num_data = X_data.shape[0]
self.inducing_variable = inducingpoint_wrapper(inducing_variable)
self.jitter_variance = Parameter(
default_jitter(), transform=positive(0.0), trainable=False, name="jitter"
)
def upper_bound(self) -> tf.Tensor:
"""
Upper bound for the sparse GP regression marginal likelihood. 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
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}
}
The key quantity, the trace term, can be computed via
>>> _, v = conditionals.conditional(X, model.inducing_variable.Z, model.kernel,
... np.zeros((len(model.inducing_variable), 1)))
which computes each individual element of the trace term.
"""
X_data, Y_data = self.data
num_data = to_default_float(tf.shape(Y_data)[0])
Kdiag = self.kernel(X_data, full_cov=False)
kuu = Kuu(self.inducing_variable, self.kernel, jitter=self.jitter_variance)
kuf = Kuf(self.inducing_variable, self.kernel, X_data)
I = tf.eye(tf.shape(kuu)[0], dtype=default_float())
L = tf.linalg.cholesky(kuu)
A = tf.linalg.triangular_solve(L, kuf, lower=True)
AAT = tf.linalg.matmul(A, A, transpose_b=True)
B = I + AAT / self.likelihood.variance
LB = tf.linalg.cholesky(B)
# Using the Trace bound, from Titsias' presentation
c = tf.reduce_sum(Kdiag) - tf.reduce_sum(tf.square(A))
# Alternative bound on max eigenval:
corrected_noise = self.likelihood.variance + c
const = -0.5 * num_data * tf.math.log(2 * np.pi * self.likelihood.variance)
logdet = -tf.reduce_sum(tf.math.log(tf.linalg.diag_part(LB)))
LC = tf.linalg.cholesky(I + AAT / corrected_noise)
v = tf.linalg.triangular_solve(
LC, tf.linalg.matmul(A, Y_data) / corrected_noise, lower=True
)
quad = -0.5 * tf.reduce_sum(tf.square(Y_data)) / corrected_noise + 0.5 * tf.reduce_sum(
tf.square(v)
)
return const + logdet + quad
class SGPR(SGPRBase):
"""
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 maximum_log_likelihood_objective(self) -> tf.Tensor:
return self.elbo()
def elbo(self) -> tf.Tensor:
"""
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.
"""
X_data, Y_data = self.data
num_inducing = len(self.inducing_variable)
num_data = to_default_float(tf.shape(Y_data)[0])
output_dim = to_default_float(tf.shape(Y_data)[1])
err = Y_data - self.mean_function(X_data)
Kdiag = self.kernel(X_data, full_cov=False)
kuf = Kuf(self.inducing_variable, self.kernel, X_data)
kuu = Kuu(self.inducing_variable, self.kernel, jitter=self.jitter_variance)
L = tf.linalg.cholesky(kuu)
sigma = tf.sqrt(self.likelihood.variance)
# Compute intermediate matrices
A = tf.linalg.triangular_solve(L, kuf, lower=True) / sigma
AAT = tf.linalg.matmul(A, A, transpose_b=True)
B = AAT + tf.eye(num_inducing, dtype=default_float())
LB = tf.linalg.cholesky(B)
Aerr = tf.linalg.matmul(A, err)
c = tf.linalg.triangular_solve(LB, Aerr, lower=True) / sigma
# compute log marginal bound
bound = -0.5 * num_data * output_dim * np.log(2 * np.pi)
bound += tf.negative(output_dim) * tf.reduce_sum(tf.math.log(tf.linalg.diag_part(LB)))
bound -= 0.5 * num_data * output_dim * tf.math.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.linalg.diag_part(AAT))
return bound
def predict_f(self, Xnew: InputData, full_cov=False, full_output_cov=False) -> MeanAndVariance:
"""
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.
"""
X_data, Y_data = self.data
num_inducing = len(self.inducing_variable)
err = Y_data - self.mean_function(X_data)
kuf = Kuf(self.inducing_variable, self.kernel, X_data)
kuu = Kuu(self.inducing_variable, self.kernel, jitter=self.jitter_variance)
Kus = Kuf(self.inducing_variable, self.kernel, Xnew)
sigma = tf.sqrt(self.likelihood.variance)
L = tf.linalg.cholesky(kuu)
A = tf.linalg.triangular_solve(L, kuf, lower=True) / sigma
B = tf.linalg.matmul(A, A, transpose_b=True) + tf.eye(num_inducing, dtype=default_float())
LB = tf.linalg.cholesky(B)
Aerr = tf.linalg.matmul(A, err)
c = tf.linalg.triangular_solve(LB, Aerr, lower=True) / sigma
tmp1 = tf.linalg.triangular_solve(L, Kus, lower=True)
tmp2 = tf.linalg.triangular_solve(LB, tmp1, lower=True)
mean = tf.linalg.matmul(tmp2, c, transpose_a=True)
if full_cov:
var = (
self.kernel(Xnew)
+ tf.linalg.matmul(tmp2, tmp2, transpose_a=True)
- tf.linalg.matmul(tmp1, tmp1, transpose_a=True)
)
var = tf.tile(var[None, ...], [self.num_latent_gps, 1, 1]) # [P, N, N]
else:
var = (
self.kernel(Xnew, full_cov=False)
+ tf.reduce_sum(tf.square(tmp2), 0)
- tf.reduce_sum(tf.square(tmp1), 0)
)
var = tf.tile(var[:, None], [1, self.num_latent_gps])
return mean + self.mean_function(Xnew), var
def compute_qu(self) -> Tuple[tf.Tensor, tf.Tensor]:
"""
Computes the mean and variance of q(u) = N(mu, cov), the variational distribution on
inducing outputs. SVGP with this q(u) should predict identically to
SGPR.
:return: mu, cov
"""
X_data, Y_data = self.data
kuf = Kuf(self.inducing_variable, self.kernel, X_data)
kuu = Kuu(self.inducing_variable, self.kernel, jitter=self.jitter_variance)
sig = kuu + (self.likelihood.variance ** -1) * tf.matmul(kuf, kuf, transpose_b=True)
sig_sqrt = tf.linalg.cholesky(sig)
sig_sqrt_kuu = tf.linalg.triangular_solve(sig_sqrt, kuu)
cov = tf.linalg.matmul(sig_sqrt_kuu, sig_sqrt_kuu, transpose_a=True)
err = Y_data - self.mean_function(X_data)
mu = (
tf.linalg.matmul(
sig_sqrt_kuu,
tf.linalg.triangular_solve(sig_sqrt, tf.linalg.matmul(kuf, err)),
transpose_a=True,
)
/ self.likelihood.variance
)
return mu, cov
class GPRFITC(SGPRBase):
"""
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.
"""
def common_terms(self):
X_data, Y_data = self.data
num_inducing = len(self.inducing_variable)
err = Y_data - self.mean_function(X_data) # size [N, R]
Kdiag = self.kernel(X_data, full_cov=False)
kuf = Kuf(self.inducing_variable, self.kernel, X_data)
kuu = Kuu(self.inducing_variable, self.kernel, jitter=self.jitter_variance)
Luu = tf.linalg.cholesky(kuu) # => Luu Luu^T = kuu
V = tf.linalg.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=default_float()) + tf.linalg.matmul(
V / nu, V, transpose_b=True
)
L = tf.linalg.cholesky(B)
beta = err / tf.expand_dims(nu, 1) # size [N, R]
alpha = tf.linalg.matmul(V, beta) # size [N, R]
gamma = tf.linalg.triangular_solve(L, alpha, lower=True) # size [N, R]
return err, nu, Luu, L, alpha, beta, gamma
def maximum_log_likelihood_objective(self) -> tf.Tensor:
return self.fitc_log_marginal_likelihood()
def fitc_log_marginal_likelihood(self) -> tf.Tensor:
"""
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.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.math.log(tf.constant(2.0 * np.pi, default_float()))
logDeterminantTerm = -0.5 * tf.reduce_sum(tf.math.log(nu)) - tf.reduce_sum(
tf.math.log(tf.linalg.diag_part(L))
)
logNormalizingTerm = constantTerm + logDeterminantTerm
return mahalanobisTerm + logNormalizingTerm * self.num_latent_gps
def predict_f(self, Xnew: InputData, full_cov=False, full_output_cov=False) -> MeanAndVariance:
"""
Compute the mean and variance of the latent function at some new points
Xnew.
"""
_, _, Luu, L, _, _, gamma = self.common_terms()
Kus = Kuf(self.inducing_variable, self.kernel, Xnew) # [M, N]
w = tf.linalg.triangular_solve(Luu, Kus, lower=True) # [M, N]
tmp = tf.linalg.triangular_solve(tf.transpose(L), gamma, lower=False)
mean = tf.linalg.matmul(w, tmp, transpose_a=True) + self.mean_function(Xnew)
intermediateA = tf.linalg.triangular_solve(L, w, lower=True)
if full_cov:
var = (
self.kernel(Xnew)
- tf.linalg.matmul(w, w, transpose_a=True)
+ tf.linalg.matmul(intermediateA, intermediateA, transpose_a=True)
)
var = tf.tile(var[None, ...], [self.num_latent_gps, 1, 1]) # [P, N, N]
else:
var = (
self.kernel(Xnew, full_cov=False)
- tf.reduce_sum(tf.square(w), 0)
+ tf.reduce_sum(tf.square(intermediateA), 0)
) # [N, P]
var = tf.tile(var[:, None], [1, self.num_latent_gps])
return mean, var