gplvm.py
``````# Copyright 2016 the GPflow authors.
#
# 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 likelihoods
from .. import kernels
from .. import features

from ..mean_functions import Zero
from ..expectations import expectation
from ..probability_distributions import DiagonalGaussian

from .model import GPModel
from .gpr import GPR

class GPLVM(GPR):
"""
Standard GPLVM where the likelihood can be optimised with respect to the latent X.
"""

def __init__(self,
Y,
latent_dim,
X_mean=None,
kernel=None,
mean_function=None,
**kwargs):
"""
Initialise GPLVM object. This method only works with a Gaussian likelihood.

:param Y: data matrix, size N (number of points) x D (dimensions)
:param Z: matrix of inducing points, size M (inducing points) x Q (latent dimensions)
:param X_mean: latent positions ([N, Q]), for the initialisation of the latent space.
:param kernel: kernel specification, by default RBF
:param mean_function: mean function, by default None.
"""
if mean_function is None:
mean_function = Zero()
if kernel is None:
kernel = kernels.RBF(latent_dim, ARD=True)
if X_mean is None:
X_mean = PCA_reduce(Y, latent_dim)
num_latent = X_mean.shape[1]
if num_latent != latent_dim:
msg = 'Passed in number of latent {0} does not match initial X {1}.'
raise ValueError(msg.format(latent_dim, num_latent))
if Y.shape[1] < num_latent:
raise ValueError('More latent dimensions than observed.')
GPR.__init__(self,
X_mean,
Y,
kernel,
mean_function=mean_function,
**kwargs)
del self.X  # in GPLVM this is a Param
self.X = Parameter(X_mean)

class BayesianGPLVM(GPModel):
def __init__(self,
X_mean,
X_var,
Y,
kernel,
M,
Z=None,
X_prior_mean=None,
X_prior_var=None):
"""
Initialise Bayesian GPLVM object. This method only works with a Gaussian likelihood.
:param X_mean: initial latent positions, size N (number of points) x Q (latent dimensions).
:param X_var: variance of latent positions ([N, Q]), for the initialisation of the latent space.
:param Y: data matrix, size N (number of points) x D (dimensions)
:param kernel: kernel specification, by default RBF
:param M: number of inducing points
:param Z: matrix of inducing points, size M (inducing points) x Q (latent dimensions). By default
random permutation of X_mean.
:param X_prior_mean: prior mean used in KL term of bound. By default 0. Same size as X_mean.
:param X_prior_var: pripor variance used in KL term of bound. By default 1.
"""
GPModel.__init__(self,
X_mean,
Y,
kernel,
likelihood=likelihoods.Gaussian(),
mean_function=Zero())
del self.X  # in GPLVM this is a Param
self.X_mean = Parameter(X_mean)
# diag_transform = transforms.DiagMatrix(X_var.shape[1])
# self.X_var() = Parameter(diag_transform.forward(transforms.positive.backward(X_var)) if X_var.ndim == 2 else X_var,
#                    diag_transform)
assert X_var.ndim == 2
self.X_var = Parameter(X_var, transform=transforms.positive)

self.num_data, self.num_latent = X_mean.shape
self.output_dim = Y.shape[1]

assert np.all(X_mean.shape == X_var.shape)
assert X_mean.shape[0] == Y.shape[0], 'X mean and Y must be same size.'
assert X_var.shape[0] == Y.shape[0], 'X var and Y must be same size.'

# inducing points
if Z is None:
# By default we initialize by subset of initial latent points
Z = np.random.permutation(X_mean.copy())[:M]

self.feature = features.InducingPoints(Z)

assert len(self.feature) == M
assert X_mean.shape[1] == self.num_latent

# deal with parameters for the prior mean variance of X
if X_prior_mean is None:
X_prior_mean = np.zeros((self.num_data, self.num_latent))
if X_prior_var is None:
X_prior_var = np.ones((self.num_data, self.num_latent))

self.X_prior_mean = np.asarray(np.atleast_1d(X_prior_mean),
dtype=default_float())
self.X_prior_var = np.asarray(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
assert self.X_prior_var.shape[0] == self.num_data
assert self.X_prior_var.shape[1] == self.num_latent

def _build_likelihood(self):
"""
Construct a tensorflow function to compute the bound on the marginal
likelihood.
"""
pX = DiagonalGaussian(self.X_mean(), self.X_var())

num_inducing = len(self.feature)
psi0 = tf.reduce_sum(expectation(pX, self.kernel))
psi1 = expectation(pX, (self.kernel, self.feature))
psi2 = tf.reduce_sum(expectation(pX, (self.kernel, self.feature),
(self.kernel, self.feature)),
axis=0)
cov_uu = Kuu(self.feature, self.kernel, jitter=default_jitter())
L = tf.linalg.cholesky(cov_uu)
sigma2 = self.likelihood.variance
sigma = tf.sqrt(sigma2)

# Compute intermediate matrices
A = tf.linalg.triangular_solve(L, tf.transpose(psi1),
lower=True) / sigma
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. * tf.reduce_sum(tf.math.log(tf.linalg.diag_part(LB)))
c = tf.linalg.triangular_solve(
LB, tf.linalg.matmul(A, self.Y), lower=True) / sigma

# KL[q(x) || p(x)]
dX_var = self.X_var() if len(
self.X_var().get_shape()) == 2 else tf.linalg.diag_part(
self.X_var())
NQ = tf.cast(tf.size(self.X_mean()), default_float())
D = tf.cast(tf.shape(self.Y)[1], default_float())
KL = -0.5 * tf.reduce_sum(tf.math.log(dX_var)) \
+ 0.5 * tf.reduce_sum(tf.math.log(self.X_prior_var)) \
- 0.5 * NQ \
+ 0.5 * tf.reduce_sum((tf.square(self.X_mean() - self.X_prior_mean) + dX_var) / self.X_prior_var)

# compute log marginal bound
ND = tf.cast(tf.size(self.Y), default_float())
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(self.Y)) / 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

def _build_predict(self, Xnew, full_cov=False):
"""
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.
:param Xnew: Point to predict at.
"""
pX = DiagonalGaussian(self.X_mean(), self.X_var())

num_inducing = len(self.feature)
psi1 = expectation(pX, (self.kernel, self.feature))
psi2 = tf.reduce_sum(expectation(pX, (self.kernel, self.feature),
(self.kernel, self.feature)),
axis=0)
jitter = default_jitter()
Kus = Kuf(self.feature, self.kernel, Xnew)
sigma2 = self.likelihood.variance
sigma = tf.sqrt(sigma2)
L = tf.linalg.cholesky(Kuu(self.feature, self.kernel, jitter=jitter))

A = tf.linalg.triangular_solve(L, tf.transpose(psi1),
lower=True) / sigma
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, self.Y), 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)
shape = tf.stack([1, 1, tf.shape(self.Y)[1]])
var = tf.tile(tf.expand_dims(var, 2), shape)
else:
var = self.kernel(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

def PCA_reduce(X, Q):
"""
A helpful function for linearly reducing the dimensionality of the data X
to Q.
:param X: data array of size N (number of points) x D (dimensions)
:param Q: Number of latent dimensions, Q < D
:return: PCA projection array of size [N, Q].
"""
assert Q <= X.shape[1], 'Cannot have more latent dimensions than observed'
evals, evecs = np.linalg.eigh(np.cov(X.T))
W = evecs[:, -Q:]
return (X - X.mean(0)).dot(W)
``````