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Tip revision: 0c3e151a3442ac8cd0eb85b9674203e75c288184 authored by Artem Artemev on 19 October 2018, 16:42:59 UTC
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conditionals.py
# Copyright 2016 Valentine Svensson, James Hensman, alexggmatthews
#
# 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.
import tensorflow as tf
from . import settings, mean_functions
from .decors import name_scope
from .dispatch import conditional, sample_conditional
from .expectations import expectation
from .features import Kuu, Kuf, InducingPoints, InducingFeature
from .kernels import Kernel, Combination
from .probability_distributions import Gaussian
logger = settings.logger()
# ----------------------------------------------------------------------------
############################### CONDITIONAL ##################################
# ----------------------------------------------------------------------------
@conditional.register(object, InducingFeature, Kernel, object)
@name_scope("conditional")
def _conditional(Xnew, feat, kern, f, *, full_cov=False, full_output_cov=False, q_sqrt=None, white=False):
"""
Single-output GP conditional.
The covariance matrices used to calculate the conditional have the following shape:
- Kuu: M x M
- Kuf: M x N
- Kff: N or N x N
Further reference
-----------------
- See `gpflow.conditionals._conditional` (below) for a detailed explanation of
conditional in the single-output case.
- See the multiouput notebook for more information about the multiouput framework.
Parameters
----------
:param Xnew: data matrix, size N x D.
:param f: data matrix, M x R
:param full_cov: return the covariance between the datapoints
:param full_output_cov: return the covariance between the outputs.
Note: as we are using a single-output kernel with repetitions these covariances will be zero.
:param q_sqrt: matrix of standard-deviations or Cholesky matrices,
size M x R or R x M x M.
:param white: boolean of whether to use the whitened representation
:return:
- mean: N x R
- variance: N x R, R x N x N, N x R x R or N x R x N x R
Please see `gpflow.conditional._expand_independent_outputs` for more information
about the shape of the variance, depending on `full_cov` and `full_output_cov`.
"""
logger.debug("Conditional: Inducing Feature - Kernel")
Kmm = Kuu(feat, kern, jitter=settings.numerics.jitter_level) # M x M
Kmn = Kuf(feat, kern, Xnew) # M x N
Knn = kern.K(Xnew) if full_cov else kern.Kdiag(Xnew)
fmean, fvar = base_conditional(Kmn, Kmm, Knn, f, full_cov=full_cov,
q_sqrt=q_sqrt, white=white) # N x R, R x N x N or N x R
return fmean, _expand_independent_outputs(fvar, full_cov, full_output_cov)
@conditional.register(object, object, Kernel, object)
@name_scope("conditional")
def _conditional(Xnew, X, kern, f, *, full_cov=False, q_sqrt=None, white=False):
"""
Given f, representing the GP at the points X, produce the mean and
(co-)variance of the GP at the points Xnew.
Additionally, there may be Gaussian uncertainty about f as represented by
q_sqrt. In this case `f` represents the mean of the distribution and
q_sqrt the square-root of the covariance.
Additionally, the GP may have been centered (whitened) so that
p(v) = N(0, I)
f = L v
thus
p(f) = N(0, LL^T) = N(0, K).
In this case `f` represents the values taken by v.
The method can either return the diagonals of the covariance matrix for
each output (default) or the full covariance matrix (full_cov=True).
We assume R independent GPs, represented by the columns of f (and the
first dimension of q_sqrt).
:param Xnew: data matrix, size N x D. Evaluate the GP at these new points
:param X: data points, size M x D.
:param kern: GPflow kernel.
:param f: data matrix, M x R, representing the function values at X,
for K functions.
:param q_sqrt: matrix of standard-deviations or Cholesky matrices,
size M x R or R x M x M.
:param white: boolean of whether to use the whitened representation as
described above.
:return:
- mean: N x R
- variance: N x R (full_cov = False), R x N x N (full_cov = True)
"""
logger.debug("Conditional: Kernel")
num_data = tf.shape(X)[0] # M
Kmm = kern.K(X) + tf.eye(num_data, dtype=settings.float_type) * settings.numerics.jitter_level
Kmn = kern.K(X, Xnew)
if full_cov:
Knn = kern.K(Xnew)
else:
Knn = kern.Kdiag(Xnew)
mean, var = base_conditional(Kmn, Kmm, Knn, f, full_cov=full_cov, q_sqrt=q_sqrt, white=white)
return mean, var # N x R, N x R or R x N x N
# ----------------------------------------------------------------------------
############################ SAMPLE CONDITIONAL ##############################
# ----------------------------------------------------------------------------
@sample_conditional.register(object, InducingFeature, Kernel, object)
@name_scope("sample_conditional")
def _sample_conditional(Xnew, feat, kern, f, *, full_cov=False, full_output_cov=False, q_sqrt=None, white=False, num_samples=None):
"""
`sample_conditional` will return a sample from the conditional distribution.
In most cases this means calculating the conditional mean m and variance v and then
returning m + sqrt(v) * eps, with eps ~ N(0, 1).
However, for some combinations of Mok and Mof more efficient sampling routines exists.
The dispatcher will make sure that we use the most efficient one.
:return: samples, mean, cov
samples has shape [num_samples, N, P] or [N, P] if num_samples is None
mean and cov as for conditional()
"""
if full_cov and full_output_cov:
raise NotImplementedError("The combination of both full_cov and full_output_cov is not "
"implemented for sample_conditional.")
logger.debug("sample conditional: InducingFeature Kernel")
mean, cov = conditional(Xnew, feat, kern, f, q_sqrt=q_sqrt, white=white,
full_cov=full_cov, full_output_cov=full_output_cov)
if full_cov:
# mean: N x P
# cov: P x N x N
mean = tf.matrix_transpose(mean) # now P x N
samples = _sample_mvn(mean, cov, 'full', num_samples=num_samples) # (S x) P x N
samples = tf.matrix_transpose(samples) # now (S x) N x P
else:
cov_structure = "full" if full_output_cov else "diag"
samples = _sample_mvn(mean, cov, cov_structure, num_samples=num_samples) # [(S,), N, P]
return samples, mean, cov
@sample_conditional.register(object, object, Kernel, object)
@name_scope("sample_conditional")
def _sample_conditional(Xnew, X, kern, f, *, q_sqrt=None, white=False, full_cov=False, full_output_cov=False, num_samples=None):
if full_cov and full_output_cov:
raise NotImplementedError("The combination of both full_cov and full_output_cov is not "
"implemented for sample_conditional.")
logger.debug("sample conditional: Kernel")
if full_output_cov:
raise NotImplementedError("full_output_cov is not implemented")
mean, cov = conditional(Xnew, X, kern, f, q_sqrt=q_sqrt, white=white, full_cov=full_cov)
if full_cov:
mean = tf.matrix_transpose(mean)
cov_structure = "full" if full_cov else "diag"
samples = _sample_mvn(mean, cov, cov_structure, num_samples=num_samples)
if full_cov:
samples = tf.matrix_transpose(samples)
return samples, mean, cov
# ----------------------------------------------------------------------------
############################# CONDITIONAL MATHS ##############################
# ----------------------------------------------------------------------------
@name_scope()
def base_conditional(Kmn, Kmm, Knn, f, *, full_cov=False, q_sqrt=None, white=False):
"""
Given a g1 and g2, and distribution p and q such that
p(g2) = N(g2;0,Kmm)
p(g1) = N(g1;0,Knn)
p(g1|g2) = N(g1;0,Knm)
And
q(g2) = N(g2;f,q_sqrt*q_sqrt^T)
This method computes the mean and (co)variance of
q(g1) = \int q(g2) p(g1|g2)
:param Kmn: M x N
:param Kmm: M x M
:param Knn: N x N or N
:param f: M x R
:param full_cov: bool
:param q_sqrt: None or R x M x M (lower triangular)
:param white: bool
:return: N x R or R x N x N
"""
logger.debug("base conditional")
# compute kernel stuff
num_func = tf.shape(f)[1] # R
Lm = tf.cholesky(Kmm)
# Compute the projection matrix A
A = tf.matrix_triangular_solve(Lm, Kmn, lower=True)
# compute the covariance due to the conditioning
if full_cov:
fvar = Knn - tf.matmul(A, A, transpose_a=True)
fvar = tf.tile(fvar[None, :, :], [num_func, 1, 1]) # R x N x N
else:
fvar = Knn - tf.reduce_sum(tf.square(A), 0)
fvar = tf.tile(fvar[None, :], [num_func, 1]) # R x N
# another backsubstitution in the unwhitened case
if not white:
A = tf.matrix_triangular_solve(tf.transpose(Lm), A, lower=False)
# construct the conditional mean
fmean = tf.matmul(A, f, transpose_a=True)
if q_sqrt is not None:
if q_sqrt.get_shape().ndims == 2:
LTA = A * tf.expand_dims(tf.transpose(q_sqrt), 2) # R x M x N
elif q_sqrt.get_shape().ndims == 3:
L = q_sqrt
A_tiled = tf.tile(tf.expand_dims(A, 0), tf.stack([num_func, 1, 1]))
LTA = tf.matmul(L, A_tiled, transpose_a=True) # R x M x N
else: # pragma: no cover
raise ValueError("Bad dimension for q_sqrt: %s" %
str(q_sqrt.get_shape().ndims))
if full_cov:
fvar = fvar + tf.matmul(LTA, LTA, transpose_a=True) # R x N x N
else:
fvar = fvar + tf.reduce_sum(tf.square(LTA), 1) # R x N
if not full_cov:
fvar = tf.transpose(fvar) # N x R
return fmean, fvar # N x R, R x N x N or N x R
# ----------------------------------------------------------------------------
############################ UNCERTAIN CONDITIONAL ###########################
# ----------------------------------------------------------------------------
@name_scope()
def uncertain_conditional(Xnew_mu, Xnew_var, feat, kern, q_mu, q_sqrt, *,
mean_function=None, full_output_cov=False, full_cov=False, white=False):
"""
Calculates the conditional for uncertain inputs Xnew, p(Xnew) = N(Xnew_mu, Xnew_var).
See ``conditional`` documentation for further reference.
:param Xnew_mu: mean of the inputs, size N x Din
:param Xnew_var: covariance matrix of the inputs, size N x Din x Din
:param feat: gpflow.InducingFeature object, only InducingPoints is supported
:param kern: gpflow kernel or ekernel object.
:param q_mu: mean inducing points, size M x Dout
:param q_sqrt: cholesky of the covariance matrix of the inducing points, size Dout x M x M
:param full_output_cov: boolean wheter to compute covariance between output dimension.
Influences the shape of return value ``fvar``. Default is False
:param white: boolean whether to use whitened representation. Default is False.
:return fmean, fvar: mean and covariance of the conditional, size ``fmean`` is N x Dout,
size ``fvar`` depends on ``full_output_cov``: if True ``f_var`` is N x Dout x Dout,
if False then ``f_var`` is N x Dout
"""
# TODO(VD): Tensorflow 1.7 doesn't support broadcasting in``tf.matmul`` and
# ``tf.matrix_triangular_solve``. This is reported in issue 216.
# As a temporary workaround, we are using ``tf.einsum`` for the matrix
# multiplications and tiling in the triangular solves.
# The code that should be used once the bug is resolved is added in comments.
if not isinstance(feat, InducingPoints):
raise NotImplementedError
if full_cov:
# TODO(VD): ``full_cov`` True would return a ``fvar`` of shape N x N x D x D,
# encoding the covariance between input datapoints as well.
# This is not implemented as this feature is only used for plotting purposes.
raise NotImplementedError
pXnew = Gaussian(Xnew_mu, Xnew_var)
num_data = tf.shape(Xnew_mu)[0] # number of new inputs (N)
num_ind = tf.shape(q_mu)[0] # number of inducing points (M)
num_func = tf.shape(q_mu)[1] # output dimension (D)
q_sqrt_r = tf.matrix_band_part(q_sqrt, -1, 0) # D x M x M
eKuf = tf.transpose(expectation(pXnew, (kern, feat))) # M x N (psi1)
Kuu = feat.Kuu(kern, jitter=settings.numerics.jitter_level) # M x M
Luu = tf.cholesky(Kuu) # M x M
if not white:
q_mu = tf.matrix_triangular_solve(Luu, q_mu, lower=True)
Luu_tiled = tf.tile(Luu[None, :, :], [num_func, 1, 1]) # remove line once issue 216 is fixed
q_sqrt_r = tf.matrix_triangular_solve(Luu_tiled, q_sqrt_r, lower=True)
Li_eKuf = tf.matrix_triangular_solve(Luu, eKuf, lower=True) # M x N
fmean = tf.matmul(Li_eKuf, q_mu, transpose_a=True)
eKff = expectation(pXnew, kern) # N (psi0)
eKuffu = expectation(pXnew, (kern, feat), (kern, feat)) # N x M x M (psi2)
Luu_tiled = tf.tile(Luu[None, :, :], [num_data, 1, 1]) # remove this line, once issue 216 is fixed
Li_eKuffu = tf.matrix_triangular_solve(Luu_tiled, eKuffu, lower=True)
Li_eKuffu_Lit = tf.matrix_triangular_solve(Luu_tiled, tf.matrix_transpose(Li_eKuffu), lower=True) # N x M x M
cov = tf.matmul(q_sqrt_r, q_sqrt_r, transpose_b=True) # D x M x M
if mean_function is None or isinstance(mean_function, mean_functions.Zero):
e_related_to_mean = tf.zeros((num_data, num_func, num_func), dtype=settings.float_type)
else:
# Update mean: \mu(x) + m(x)
fmean = fmean + expectation(pXnew, mean_function)
# Calculate: m(x) m(x)^T + m(x) \mu(x)^T + \mu(x) m(x)^T,
# where m(x) is the mean_function and \mu(x) is fmean
e_mean_mean = expectation(pXnew, mean_function, mean_function) # N x D x D
Lit_q_mu = tf.matrix_triangular_solve(Luu, q_mu, adjoint=True)
e_mean_Kuf = expectation(pXnew, mean_function, (kern, feat)) # N x D x M
# einsum isn't able to infer the rank of e_mean_Kuf, hence we explicitly set the rank of the tensor:
e_mean_Kuf = tf.reshape(e_mean_Kuf, [num_data, num_func, num_ind])
e_fmean_mean = tf.einsum("nqm,mz->nqz", e_mean_Kuf, Lit_q_mu) # N x D x D
e_related_to_mean = e_fmean_mean + tf.matrix_transpose(e_fmean_mean) + e_mean_mean
if full_output_cov:
fvar = (
tf.matrix_diag(tf.tile((eKff - tf.trace(Li_eKuffu_Lit))[:, None], [1, num_func])) +
tf.matrix_diag(tf.einsum("nij,dji->nd", Li_eKuffu_Lit, cov)) +
# tf.matrix_diag(tf.trace(tf.matmul(Li_eKuffu_Lit, cov))) +
tf.einsum("ig,nij,jh->ngh", q_mu, Li_eKuffu_Lit, q_mu) -
# tf.matmul(q_mu, tf.matmul(Li_eKuffu_Lit, q_mu), transpose_a=True) -
fmean[:, :, None] * fmean[:, None, :] +
e_related_to_mean
)
else:
fvar = (
(eKff - tf.trace(Li_eKuffu_Lit))[:, None] +
tf.einsum("nij,dji->nd", Li_eKuffu_Lit, cov) +
tf.einsum("ig,nij,jg->ng", q_mu, Li_eKuffu_Lit, q_mu) -
fmean ** 2 +
tf.matrix_diag_part(e_related_to_mean)
)
return fmean, fvar
# ---------------------------------------------------------------
########################## HELPERS ##############################
# ---------------------------------------------------------------
def _sample_mvn(mean, cov, cov_structure=None, num_samples=None):
"""
Returns a sample from a D-dimensional Multivariate Normal distribution
:param mean: N x D
:param cov: N x D or N x D x D
:param cov_structure: "diag" or "full"
- "diag": cov holds the diagonal elements of the covariance matrix
- "full": cov holds the full covariance matrix (without jitter)
:return: sample from the MVN of shape N x D
"""
mean_shape = tf.shape(mean)
cov_shape = tf.shape(cov)
N, D = mean_shape[0], mean_shape[1]
S = num_samples if num_samples is not None else 1
# assert shape(cov) == (N, D) or (N, D, D)
with tf.control_dependencies([
tf.Assert(tf.equal(cov_shape[0], N) & tf.reduce_all(tf.equal(cov_shape[1:], D)),
data=[mean_shape, cov_shape])
]):
if cov_structure == "diag":
with tf.control_dependencies([tf.assert_equal(tf.rank(mean), tf.rank(cov))]):
eps = tf.random_normal([S, N, D], dtype=settings.float_type) # S x N x D
samples = mean + tf.sqrt(cov) * eps # S x N x D
elif cov_structure == "full":
with tf.control_dependencies([tf.assert_equal(tf.rank(mean) + 1, tf.rank(cov))]):
jittermat = settings.numerics.jitter_level * \
tf.eye(D, batch_shape=[N], dtype=settings.float_type) # N x D x D
eps = tf.random_normal([N, D, S], dtype=settings.float_type) # N x D x S
chol = tf.cholesky(cov + jittermat) # N x D x D
samples = mean[..., None] + tf.matmul(chol, eps) # N x D x S
samples = tf.transpose(samples, [2, 0, 1]) # S x N x D
else:
raise NotImplementedError # pragma: no cover
if num_samples is None:
return samples[0] # N x D
return samples # S x N x D
def _expand_independent_outputs(fvar, full_cov, full_output_cov):
"""
Reshapes fvar to the correct shape, specified by `full_cov` and `full_output_cov`.
:param fvar: has shape N x P (full_cov = False) or P x N x N (full_cov = True).
:return:
1. full_cov: True and full_output_cov: True
fvar N x P x N x P
2. full_cov: True and full_output_cov: False
fvar P x N x N
3. full_cov: False and full_output_cov: True
fvar N x P x P
4. full_cov: False and full_output_cov: False
fvar N x P
"""
if full_cov and full_output_cov:
fvar = tf.matrix_diag(tf.transpose(fvar)) # N x N x P x P
fvar = tf.transpose(fvar, [0, 2, 1, 3]) # N x P x N x P
if not full_cov and full_output_cov:
fvar = tf.matrix_diag(fvar) # N x P x P
if full_cov and not full_output_cov:
pass # P x N x N
if not full_cov and not full_output_cov:
pass # N x P
return fvar