# Copyright 2016-2020 The GPflow Contributors. All Rights Reserved.
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# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
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# http://www.apache.org/licenses/LICENSE-2.0
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from typing import Optional
import numpy as np
import tensorflow as tf
from .. import covariances, kernels, likelihoods
from ..base import Parameter
from ..config import default_float, default_jitter
from ..expectations import expectation
from ..inducing_variables import InducingPoints
from ..kernels import Kernel
from ..mean_functions import MeanFunction, Zero
from ..probability_distributions import DiagonalGaussian
from ..utilities import positive, to_default_float
from ..utilities.ops import pca_reduce
from .gpr import GPR
from .model import GPModel, MeanAndVariance
from .training_mixins import InputData, InternalDataTrainingLossMixin, OutputData
from .util import data_input_to_tensor, inducingpoint_wrapper
[docs]class GPLVM(GPR):
"""
Standard GPLVM where the likelihood can be optimised with respect to the latent X.
"""
def __init__(
self,
data: OutputData,
latent_dim: int,
X_data_mean: Optional[tf.Tensor] = None,
kernel: Optional[Kernel] = None,
mean_function: Optional[MeanFunction] = None,
):
"""
Initialise GPLVM object. This method only works with a Gaussian likelihood.
:param data: y data matrix, size N (number of points) x D (dimensions)
:param latent_dim: the number of latent dimensions (Q)
:param X_data_mean: latent positions ([N, Q]), for the initialisation of the latent space.
:param kernel: kernel specification, by default Squared Exponential
:param mean_function: mean function, by default None.
"""
if X_data_mean is None:
X_data_mean = pca_reduce(data, latent_dim)
num_latent_gps = X_data_mean.shape[1]
if num_latent_gps != latent_dim:
msg = "Passed in number of latent {0} does not match initial X {1}."
raise ValueError(msg.format(latent_dim, num_latent_gps))
if mean_function is None:
mean_function = Zero()
if kernel is None:
kernel = kernels.SquaredExponential(lengthscales=tf.ones((latent_dim,)))
if data.shape[1] < num_latent_gps:
raise ValueError("More latent dimensions than observed.")
gpr_data = (Parameter(X_data_mean), data_input_to_tensor(data))
super().__init__(gpr_data, kernel, mean_function=mean_function)
[docs]class BayesianGPLVM(GPModel, InternalDataTrainingLossMixin):
def __init__(
self,
data: OutputData,
X_data_mean: tf.Tensor,
X_data_var: tf.Tensor,
kernel: Kernel,
num_inducing_variables: Optional[int] = None,
inducing_variable=None,
X_prior_mean=None,
X_prior_var=None,
):
"""
Initialise Bayesian GPLVM object. This method only works with a Gaussian likelihood.
:param data: data matrix, size N (number of points) x D (dimensions)
:param X_data_mean: initial latent positions, size N (number of points) x Q (latent dimensions).
:param X_data_var: variance of latent positions ([N, Q]), for the initialisation of the latent space.
:param kernel: kernel specification, by default Squared Exponential
:param num_inducing_variables: number of inducing points, M
:param inducing_variable: matrix of inducing points, size M (inducing points) x Q (latent dimensions). By default
random permutation of X_data_mean.
:param X_prior_mean: prior mean used in KL term of bound. By default 0. Same size as X_data_mean.
:param X_prior_var: prior variance used in KL term of bound. By default 1.
"""
num_data, num_latent_gps = X_data_mean.shape
super().__init__(kernel, likelihoods.Gaussian(), num_latent_gps=num_latent_gps)
self.data = data_input_to_tensor(data)
assert X_data_var.ndim == 2
self.X_data_mean = Parameter(X_data_mean)
self.X_data_var = Parameter(X_data_var, transform=positive())
self.num_data = num_data
self.output_dim = self.data.shape[-1]
assert np.all(X_data_mean.shape == X_data_var.shape)
assert X_data_mean.shape[0] == self.data.shape[0], "X mean and Y must be same size."
assert X_data_var.shape[0] == self.data.shape[0], "X var and Y must be same size."
if (inducing_variable is None) == (num_inducing_variables is None):
raise ValueError(
"BayesianGPLVM needs exactly one of `inducing_variable` and `num_inducing_variables`"
)
if inducing_variable is None:
# By default we initialize by subset of initial latent points
# Note that tf.random.shuffle returns a copy, it does not shuffle in-place
Z = tf.random.shuffle(X_data_mean)[:num_inducing_variables]
inducing_variable = InducingPoints(Z)
self.inducing_variable = inducingpoint_wrapper(inducing_variable)
assert X_data_mean.shape[1] == self.num_latent_gps
# deal with parameters for the prior mean variance of X
if X_prior_mean is None:
X_prior_mean = tf.zeros((self.num_data, self.num_latent_gps), dtype=default_float())
if X_prior_var is None:
X_prior_var = tf.ones((self.num_data, self.num_latent_gps))
self.X_prior_mean = tf.convert_to_tensor(np.atleast_1d(X_prior_mean), dtype=default_float())
self.X_prior_var = tf.convert_to_tensor(np.atleast_1d(X_prior_var), dtype=default_float())
assert self.X_prior_mean.shape[0] == self.num_data
assert self.X_prior_mean.shape[1] == self.num_latent_gps
assert self.X_prior_var.shape[0] == self.num_data
assert self.X_prior_var.shape[1] == self.num_latent_gps
[docs] def maximum_log_likelihood_objective(self) -> tf.Tensor:
return self.elbo()
[docs] def elbo(self) -> tf.Tensor:
"""
Construct a tensorflow function to compute the bound on the marginal
likelihood.
"""
Y_data = self.data
pX = DiagonalGaussian(self.X_data_mean, self.X_data_var)
num_inducing = self.inducing_variable.num_inducing
psi0 = tf.reduce_sum(expectation(pX, self.kernel))
psi1 = expectation(pX, (self.kernel, self.inducing_variable))
psi2 = tf.reduce_sum(
expectation(
pX, (self.kernel, self.inducing_variable), (self.kernel, self.inducing_variable)
),
axis=0,
)
cov_uu = covariances.Kuu(self.inducing_variable, self.kernel, jitter=default_jitter())
L = tf.linalg.cholesky(cov_uu)
sigma2 = self.likelihood.variance
# Compute intermediate matrices
A = tf.linalg.triangular_solve(L, tf.transpose(psi1), lower=True)
tmp = tf.linalg.triangular_solve(L, psi2, lower=True)
AAT = tf.linalg.triangular_solve(L, tf.transpose(tmp), lower=True) / sigma2
B = AAT + tf.eye(num_inducing, dtype=default_float())
LB = tf.linalg.cholesky(B)
log_det_B = 2.0 * tf.reduce_sum(tf.math.log(tf.linalg.diag_part(LB)))
c = tf.linalg.triangular_solve(LB, tf.linalg.matmul(A, Y_data), lower=True) / sigma2
# KL[q(x) || p(x)]
dX_data_var = (
self.X_data_var
if self.X_data_var.shape.ndims == 2
else tf.linalg.diag_part(self.X_data_var)
)
NQ = to_default_float(tf.size(self.X_data_mean))
D = to_default_float(tf.shape(Y_data)[1])
KL = -0.5 * tf.reduce_sum(tf.math.log(dX_data_var))
KL += 0.5 * tf.reduce_sum(tf.math.log(self.X_prior_var))
KL -= 0.5 * NQ
KL += 0.5 * tf.reduce_sum(
(tf.square(self.X_data_mean - self.X_prior_mean) + dX_data_var) / self.X_prior_var
)
# compute log marginal bound
ND = to_default_float(tf.size(Y_data))
bound = -0.5 * ND * tf.math.log(2 * np.pi * sigma2)
bound += -0.5 * D * log_det_B
bound += -0.5 * tf.reduce_sum(tf.square(Y_data)) / sigma2
bound += 0.5 * tf.reduce_sum(tf.square(c))
bound += -0.5 * D * (tf.reduce_sum(psi0) / sigma2 - tf.reduce_sum(tf.linalg.diag_part(AAT)))
bound -= KL
return bound
[docs] def predict_f(
self, Xnew: InputData, full_cov: bool = False, full_output_cov: bool = False
) -> MeanAndVariance:
"""
Compute the mean and variance of the latent function at some new points.
Note that this is very similar to the SGPR prediction, for which
there are notes in the SGPR notebook.
Note: This model does not allow full output covariances.
:param Xnew: points at which to predict
"""
if full_output_cov:
raise NotImplementedError
pX = DiagonalGaussian(self.X_data_mean, self.X_data_var)
Y_data = self.data
num_inducing = self.inducing_variable.num_inducing
psi1 = expectation(pX, (self.kernel, self.inducing_variable))
psi2 = tf.reduce_sum(
expectation(
pX, (self.kernel, self.inducing_variable), (self.kernel, self.inducing_variable)
),
axis=0,
)
jitter = default_jitter()
Kus = covariances.Kuf(self.inducing_variable, self.kernel, Xnew)
sigma2 = self.likelihood.variance
L = tf.linalg.cholesky(covariances.Kuu(self.inducing_variable, self.kernel, jitter=jitter))
A = tf.linalg.triangular_solve(L, tf.transpose(psi1), lower=True)
tmp = tf.linalg.triangular_solve(L, psi2, lower=True)
AAT = tf.linalg.triangular_solve(L, tf.transpose(tmp), lower=True) / sigma2
B = AAT + tf.eye(num_inducing, dtype=default_float())
LB = tf.linalg.cholesky(B)
c = tf.linalg.triangular_solve(LB, tf.linalg.matmul(A, Y_data), lower=True) / sigma2
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)
)
shape = tf.stack([1, 1, tf.shape(Y_data)[1]])
var = tf.tile(tf.expand_dims(var, 2), shape)
else:
var = (
self.kernel(Xnew, full_cov=False)
+ tf.reduce_sum(tf.square(tmp2), axis=0)
- tf.reduce_sum(tf.square(tmp1), axis=0)
)
shape = tf.stack([1, tf.shape(Y_data)[1]])
var = tf.tile(tf.expand_dims(var, 1), shape)
return mean + self.mean_function(Xnew), var
[docs] def predict_log_density(self, data: OutputData) -> tf.Tensor:
raise NotImplementedError