# Copyright 2016-2020 The GPflow Contributors. All Rights Reserved.
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# Licensed under the Apache License, Version 2.0 (the "License");
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from typing import Optional
import numpy as np
import tensorflow as tf
import gpflow
from ..base import InputData, MeanAndVariance, Parameter, RegressionData
from ..conditionals import conditional
from ..config import default_float, default_jitter
from ..kernels import Kernel
from ..kullback_leiblers import gauss_kl
from ..likelihoods import Likelihood
from ..mean_functions import MeanFunction, Zero
from ..utilities import triangular
from .model import GPModel
from .training_mixins import InternalDataTrainingLossMixin
from .util import data_input_to_tensor, inducingpoint_wrapper
class VGP(GPModel, InternalDataTrainingLossMixin):
r"""
This method approximates the Gaussian process posterior using a multivariate Gaussian.
The idea is that the posterior over the function-value vector F is
approximated by a Gaussian, and the KL divergence is minimised between
the approximation and the posterior.
This implementation is equivalent to SVGP with X=Z, but is more efficient.
The whitened representation is used to aid optimization.
The posterior approximation is
.. math::
q(\mathbf f) = N(\mathbf f \,|\, \boldsymbol \mu, \boldsymbol \Sigma)
"""
def __init__(
self,
data: RegressionData,
kernel: Kernel,
likelihood: Likelihood,
mean_function: Optional[MeanFunction] = None,
num_latent_gps: Optional[int] = None,
):
"""
data = (X, Y) contains the input points [N, D] and the observations [N, P]
kernel, likelihood, mean_function are appropriate GPflow objects
"""
if num_latent_gps is None:
num_latent_gps = self.calc_num_latent_gps_from_data(data, kernel, likelihood)
super().__init__(kernel, likelihood, mean_function, num_latent_gps)
self.data = data_input_to_tensor(data)
X_data, Y_data = self.data
num_data = X_data.shape[0]
self.num_data = num_data
self.q_mu = Parameter(np.zeros((num_data, self.num_latent_gps)))
q_sqrt = np.array([np.eye(num_data) for _ in range(self.num_latent_gps)])
self.q_sqrt = Parameter(q_sqrt, transform=triangular())
def maximum_log_likelihood_objective(self) -> tf.Tensor:
return self.elbo()
def elbo(self) -> tf.Tensor:
r"""
This method computes the variational lower bound on the likelihood,
which is:
E_{q(F)} [ \log p(Y|F) ] - KL[ q(F) || p(F)]
with
q(\mathbf f) = N(\mathbf f \,|\, \boldsymbol \mu, \boldsymbol \Sigma)
"""
X_data, Y_data = self.data
# Get prior KL.
KL = gauss_kl(self.q_mu, self.q_sqrt)
# Get conditionals
K = self.kernel(X_data) + tf.eye(self.num_data, dtype=default_float()) * default_jitter()
L = tf.linalg.cholesky(K)
fmean = tf.linalg.matmul(L, self.q_mu) + self.mean_function(X_data) # [NN, ND] -> ND
q_sqrt_dnn = tf.linalg.band_part(self.q_sqrt, -1, 0) # [D, N, N]
L_tiled = tf.tile(tf.expand_dims(L, 0), tf.stack([self.num_latent_gps, 1, 1]))
LTA = tf.linalg.matmul(L_tiled, q_sqrt_dnn) # [D, N, N]
fvar = tf.reduce_sum(tf.square(LTA), 2)
fvar = tf.transpose(fvar)
# Get variational expectations.
var_exp = self.likelihood.variational_expectations(fmean, fvar, Y_data)
return tf.reduce_sum(var_exp) - KL
def predict_f(
self, Xnew: InputData, full_cov: bool = False, full_output_cov: bool = False
) -> MeanAndVariance:
X_data, _ = self.data
mu, var = conditional(
Xnew,
X_data,
self.kernel,
self.q_mu,
q_sqrt=self.q_sqrt,
full_cov=full_cov,
white=True,
)
return mu + self.mean_function(Xnew), var
class VGPOpperArchambeau(GPModel, InternalDataTrainingLossMixin):
r"""
This method approximates the Gaussian process posterior using a multivariate Gaussian.
The key reference is:
::
@article{Opper:2009,
title = {The Variational Gaussian Approximation Revisited},
author = {Opper, Manfred and Archambeau, Cedric},
journal = {Neural Comput.},
year = {2009},
pages = {786--792},
}
The idea is that the posterior over the function-value vector F is
approximated by a Gaussian, and the KL divergence is minimised between
the approximation and the posterior. It turns out that the optimal
posterior precision shares off-diagonal elements with the prior, so
only the diagonal elements of the precision need be adjusted.
The posterior approximation is
.. math::
q(\mathbf f) = N(\mathbf f \,|\, \mathbf K \boldsymbol \alpha,
[\mathbf K^{-1} + \textrm{diag}(\boldsymbol \lambda))^2]^{-1})
This approach has only 2ND parameters, rather than the N + N^2 of vgp,
but the optimization is non-convex and in practice may cause difficulty.
"""
def __init__(
self,
data: RegressionData,
kernel: Kernel,
likelihood: Likelihood,
mean_function: Optional[MeanFunction] = None,
num_latent_gps: Optional[int] = None,
):
"""
data = (X, Y) contains the input points [N, D] and the observations [N, P]
kernel, likelihood, mean_function are appropriate GPflow objects
"""
if num_latent_gps is None:
num_latent_gps = self.calc_num_latent_gps_from_data(data, kernel, likelihood)
super().__init__(kernel, likelihood, mean_function, num_latent_gps)
self.data = data_input_to_tensor(data)
X_data, Y_data = self.data
self.num_data = X_data.shape[0]
self.q_alpha = Parameter(np.zeros((self.num_data, self.num_latent_gps)))
self.q_lambda = Parameter(
np.ones((self.num_data, self.num_latent_gps)), transform=gpflow.utilities.positive()
)
def maximum_log_likelihood_objective(self) -> tf.Tensor:
return self.elbo()
def elbo(self) -> tf.Tensor:
r"""
q_alpha, q_lambda are variational parameters, size [N, R]
This method computes the variational lower bound on the likelihood,
which is:
E_{q(F)} [ \log p(Y|F) ] - KL[ q(F) || p(F)]
with
q(f) = N(f | K alpha + mean, [K^-1 + diag(square(lambda))]^-1) .
"""
X_data, Y_data = self.data
K = self.kernel(X_data)
K_alpha = tf.linalg.matmul(K, self.q_alpha)
f_mean = K_alpha + self.mean_function(X_data)
# compute the variance for each of the outputs
I = tf.tile(
tf.eye(self.num_data, dtype=default_float())[None, ...], [self.num_latent_gps, 1, 1]
)
A = (
I
+ tf.transpose(self.q_lambda)[:, None, ...]
* tf.transpose(self.q_lambda)[:, :, None, ...]
* K
)
L = tf.linalg.cholesky(A)
Li = tf.linalg.triangular_solve(L, I)
tmp = Li / tf.transpose(self.q_lambda)[:, None, ...]
f_var = 1.0 / tf.square(self.q_lambda) - tf.transpose(tf.reduce_sum(tf.square(tmp), 1))
# some statistics about A are used in the KL
A_logdet = 2.0 * tf.reduce_sum(tf.math.log(tf.linalg.diag_part(L)))
trAi = tf.reduce_sum(tf.square(Li))
KL = 0.5 * (
A_logdet
+ trAi
- self.num_data * self.num_latent_gps
+ tf.reduce_sum(K_alpha * self.q_alpha)
)
v_exp = self.likelihood.variational_expectations(f_mean, f_var, Y_data)
return tf.reduce_sum(v_exp) - KL
def predict_f(
self, Xnew: InputData, full_cov: bool = False, full_output_cov: bool = False
) -> MeanAndVariance:
r"""
The posterior variance of F is given by
q(f) = N(f | K alpha + mean, [K^-1 + diag(lambda**2)]^-1)
Here we project this to F*, the values of the GP at Xnew which is given
by
q(F*) = N ( F* | K_{*F} alpha + mean, K_{**} - K_{*f}[K_{ff} +
diag(lambda**-2)]^-1 K_{f*} )
Note: This model currently does not allow full output covariances
"""
if full_output_cov:
raise NotImplementedError
X_data, _ = self.data
# compute kernel things
Kx = self.kernel(X_data, Xnew)
K = self.kernel(X_data)
# predictive mean
f_mean = tf.linalg.matmul(Kx, self.q_alpha, transpose_a=True) + self.mean_function(Xnew)
# predictive var
A = K + tf.linalg.diag(tf.transpose(1.0 / tf.square(self.q_lambda)))
L = tf.linalg.cholesky(A)
Kx_tiled = tf.tile(Kx[None, ...], [self.num_latent_gps, 1, 1])
LiKx = tf.linalg.triangular_solve(L, Kx_tiled)
if full_cov:
f_var = self.kernel(Xnew) - tf.linalg.matmul(LiKx, LiKx, transpose_a=True)
else:
f_var = self.kernel(Xnew, full_cov=False) - tf.reduce_sum(tf.square(LiKx), axis=1)
return f_mean, tf.transpose(f_var)