vgp.py
# Copyright 2016 James Hensman, Valentine Svensson, alexggmatthews, fujiisoup
#
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#
# Unless required by applicable law or agreed to in writing, software
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and

import tensorflow as tf
import numpy as np

from .. import settings
from .. import transforms

from ..params import Parameter
from ..params import DataHolder
from ..decors import params_as_tensors

from ..mean_functions import Zero
from ..conditionals import conditional
from ..kullback_leiblers import gauss_kl
from ..models.model import GPModel

class VGP(GPModel):
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, X, Y, kern, likelihood,
mean_function=None,
num_latent=None,
**kwargs):
"""
X is a data matrix, size N x D
Y is a data matrix, size N x R
kern, likelihood, mean_function are appropriate GPflow objects

"""

X = DataHolder(X)
Y = DataHolder(Y)
GPModel.__init__(self, X, Y, kern, likelihood, mean_function, num_latent, **kwargs)
self.num_data = X.shape[0]

self.q_mu = Parameter(np.zeros((self.num_data, self.num_latent)))
q_sqrt = np.array([np.eye(self.num_data)
for _ in range(self.num_latent)])
transform = transforms.LowerTriangular(self.num_data, self.num_latent)
self.q_sqrt = Parameter(q_sqrt, transform=transform)

def compile(self, session=None):
"""
Before calling the standard compile function, check to see if the size
of the data has changed and add variational parameters appropriately.

This is necessary because the shape of the parameters depends on the
shape of the data.
"""
if not self.num_data == self.X.shape[0]:
self.num_data = self.X.shape[0]
self.q_mu = Parameter(np.zeros((self.num_data, self.num_latent)))
self.q_sqrt = Parameter(np.eye(self.num_data)[:, :, None] *
np.ones((1, 1, self.num_latent)))

return super(VGP, self).compile(session=session)

@params_as_tensors
def _build_likelihood(self):
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)

"""

# Get prior KL.
KL = gauss_kl(self.q_mu, self.q_sqrt)

# Get conditionals
K = self.kern.K(self.X) + tf.eye(self.num_data, dtype=settings.float_type) * \
settings.numerics.jitter_level
L = tf.cholesky(K)

fmean = tf.matmul(L, self.q_mu) + self.mean_function(self.X)  # NN,ND->ND

q_sqrt_dnn = tf.matrix_band_part(self.q_sqrt, -1, 0)  # D x N x N

L_tiled = tf.tile(tf.expand_dims(L, 0), tf.stack([self.num_latent, 1, 1]))

LTA = tf.matmul(L_tiled, q_sqrt_dnn)  # D x N x N
fvar = tf.reduce_sum(tf.square(LTA), 2)

fvar = tf.transpose(fvar)

# Get variational expectations.
var_exp = self.likelihood.variational_expectations(fmean, fvar, self.Y)

return tf.reduce_sum(var_exp) - KL

@params_as_tensors
def _build_predict(self, Xnew, full_cov=False):
mu, var = conditional(Xnew, self.X, self.kern, self.q_mu,
q_sqrt=self.q_sqrt, full_cov=full_cov, white=True)
return mu + self.mean_function(Xnew), var

class VGP_opper_archambeau(GPModel):
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, X, Y, kern, likelihood,
mean_function=None,
num_latent=None,
**kwargs):
"""
X is a data matrix, size N x D
Y is a data matrix, size N x R
kern, likelihood, mean_function are appropriate GPflow objects
"""

mean_function = Zero() if mean_function is None else mean_function

X = DataHolder(X)
Y = DataHolder(Y)
GPModel.__init__(self, X, Y, kern, likelihood, mean_function, **kwargs)
self.num_data = X.shape[0]
self.num_latent = num_latent or Y.shape[1]
self.q_alpha = Parameter(np.zeros((self.num_data, self.num_latent)))
self.q_lambda = Parameter(np.ones((self.num_data, self.num_latent)),
transforms.positive)

def compile(self, session=None):
"""
Before calling the standard compile function, check to see if the size
of the data has changed and add variational parameters appropriately.

This is necessary because the shape of the parameters depends on the
shape of the data.
"""
if not self.num_data == self.X.shape[0]:
self.num_data = self.X.shape[0]
self.q_alpha = Parameter(np.zeros((self.num_data, self.num_latent)))
self.q_lambda = Parameter(np.ones((self.num_data, self.num_latent)),
transforms.positive)
return super(VGP_opper_archambeau, self).compile(session=session)

@params_as_tensors
def _build_likelihood(self):
r"""
q_alpha, q_lambda are variational parameters, size N x 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) .
"""
K = self.kern.K(self.X)
K_alpha = tf.matmul(K, self.q_alpha)
f_mean = K_alpha + self.mean_function(self.X)

# compute the variance for each of the outputs
I = tf.tile(tf.expand_dims(tf.eye(self.num_data, dtype=settings.float_type), 0),
[self.num_latent, 1, 1])
A = I + tf.expand_dims(tf.transpose(self.q_lambda), 1) * \
tf.expand_dims(tf.transpose(self.q_lambda), 2) * K
L = tf.cholesky(A)
Li = tf.matrix_triangular_solve(L, I)
tmp = Li / tf.expand_dims(tf.transpose(self.q_lambda), 1)
f_var = 1. / 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.log(tf.matrix_diag_part(L)))
trAi = tf.reduce_sum(tf.square(Li))

KL = 0.5 * (A_logdet + trAi - self.num_data * self.num_latent +
tf.reduce_sum(K_alpha * self.q_alpha))

v_exp = self.likelihood.variational_expectations(f_mean, f_var, self.Y)
return tf.reduce_sum(v_exp) - KL

@params_as_tensors
def _build_predict(self, Xnew, full_cov=False):
"""
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*} )
"""

# compute kernel things
Kx = self.kern.K(self.X, Xnew)
K = self.kern.K(self.X)

# predictive mean
f_mean = tf.matmul(Kx, self.q_alpha, transpose_a=True) + self.mean_function(Xnew)

# predictive var
A = K + tf.matrix_diag(tf.transpose(1. / tf.square(self.q_lambda)))
L = tf.cholesky(A)
Kx_tiled = tf.tile(tf.expand_dims(Kx, 0), [self.num_latent, 1, 1])
LiKx = tf.matrix_triangular_solve(L, Kx_tiled)
if full_cov:
f_var = self.kern.K(Xnew) - tf.matmul(LiKx, LiKx, transpose_a=True)
else:
f_var = self.kern.Kdiag(Xnew) - tf.reduce_sum(tf.square(LiKx), 1)
return f_mean, tf.transpose(f_var)