kernels.py
# Copyright 2018 GPflow authors
#
# 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 kernels
from .. import settings
from ..decors import params_as_tensors, autoflow
from ..kernels import Kernel, Combination
from ..params import Parameter
class Mok(Kernel):
"""
Multi Output Kernel class.
This kernel can represent correlation between outputs of different datapoints.
Therefore, subclasses of Mok should implement `K` which returns:
- N x P x N x P if full_output_cov = True
- P x N x N if full_output_cov = False
and `Kdiag` returns:
- N x P x P if full_output_cov = True
- N x P if full_output_cov = False
The `full_output_cov` argument holds whether the kernel should calculate
the covariance between the outputs. In case there is no correlation but
`full_output_cov` is set to True the covariance matrix will be filled with zeros
until the appropriate size is reached.
"""
def K(self, X, X2=None, full_output_cov=True):
"""
Returns the correlation of f(X1) and f(X2), where f(.) can be multi-dimensional.
:param X: data matrix, N1 x D
:param X2: data matrix, N2 x D
:param full_output_cov: calculate correlation between outputs.
:return: cov[f(X1), f(X2)] with shape
- N1 x P x N2 x P if `full_output_cov` = True
- P x N1 x N2 if `full_output_cov` = False
"""
raise NotImplemented # pragma: no cover
def Kdiag(self, X, full_output_cov=True):
"""
Returns the correlation of f(X) and f(X), where f(.) can be multi-dimensional.
:param X: data matrix, N x D
:param full_output_cov: calculate correlation between outputs.
:return: var[f(X)] with shape
- N x P x N x P if `full_output_cov` = True
- N x P if `full_output_cov` = False
"""
raise NotImplemented # pragma: no cover
class SharedIndependentMok(Mok):
"""
- Shared: we use the same kernel for each latent GP
- Independent: Latents are uncorrelated a priori.
Note: this class is created only for testing and comparison purposes.
Use `gpflow.kernels` instead for more efficient code.
"""
def __init__(self, kern: Kernel, output_dimensionality, name=None):
Mok.__init__(self, kern.input_dim, name)
self.kern = kern
self.P = output_dimensionality
def K(self, X, X2=None, full_output_cov=True):
K = self.kern.K(X, X2) # N x N2
if full_output_cov:
Ks = tf.tile(K[..., None], [1, 1, self.P]) # N x N2 x P
return tf.transpose(tf.matrix_diag(Ks), [0, 2, 1, 3]) # N x P x N2 x P
else:
return tf.tile(K[None, ...], [self.P, 1, 1]) # P x N x N2
def Kdiag(self, X, full_output_cov=True):
K = self.kern.Kdiag(X) # N
Ks = tf.tile(K[:, None], [1, self.P]) # N x P
return tf.matrix_diag(Ks) if full_output_cov else Ks # N x P x P or N x P
class SeparateIndependentMok(Mok, Combination):
"""
- Separate: we use different kernel for each output latent
- Independent: Latents are uncorrelated a priori.
"""
def __init__(self, kernels, name=None):
Combination.__init__(self, kernels, name)
def K(self, X, X2=None, full_output_cov=True):
if full_output_cov:
Kxxs = tf.stack([k.K(X, X2) for k in self.kernels], axis=2) # N x N2 x P
return tf.transpose(tf.matrix_diag(Kxxs), [0, 2, 1, 3]) # N x P x N2 x P
else:
return tf.stack([k.K(X, X2) for k in self.kernels], axis=0) # P x N x N2
def Kdiag(self, X, full_output_cov=False):
stacked = tf.stack([k.Kdiag(X) for k in self.kernels], axis=1) # N x P
return tf.matrix_diag(stacked) if full_output_cov else stacked # N x P x P or N x P
class SeparateMixedMok(Mok, Combination):
"""
Linear mixing of the latent GPs to form the output
"""
def __init__(self, kernels, W, name=None):
Combination.__init__(self, kernels, name)
self.W = Parameter(W) # P x L
@params_as_tensors
def Kgg(self, X, X2):
return tf.stack([k.K(X, X2) for k in self.kernels], axis=0) # L x N x N2
@autoflow((settings.float_type, [None, None]),
(settings.float_type, [None, None]))
def compute_Kgg(self, X, X2):
return self.Kgg(X, X2)
@params_as_tensors
def K(self, X, X2=None, full_output_cov=True):
Kxx = self.Kgg(X, X2) # L x N x N2
KxxW = Kxx[None, :, :, :] * self.W[:, :, None, None] # P x L x N x N2
if full_output_cov:
# return tf.einsum('lnm,kl,ql->nkmq', Kxx, self.W, self.W)
WKxxW = tf.tensordot(self.W, KxxW, [[1], [1]]) # P x P x N x N2
return tf.transpose(WKxxW, [2, 0, 3, 1]) # N x P x N2 x P
else:
# return tf.einsum('lnm,kl,kl->knm', Kxx, self.W, self.W)
return tf.reduce_sum(self.W[:, :, None, None] * KxxW, [1]) # P x N x N2
@params_as_tensors
def Kdiag(self, X, full_output_cov=True):
K = tf.stack([k.Kdiag(X) for k in self.kernels], axis=1) # N x L
if full_output_cov:
# Can currently not use einsum due to unknown shape from `tf.stack()`
# return tf.einsum('nl,lk,lq->nkq', K, self.W, self.W) # N x P x P
Wt = tf.transpose(self.W) # L x P
return tf.reduce_sum(K[:, :, None, None] * Wt[None, :, :, None] * Wt[None, :, None, :], axis=1) # N x P x P
else:
# return tf.einsum('nl,lk,lk->nkq', K, self.W, self.W) # N x P
return tf.matmul(K, self.W ** 2.0, transpose_b=True) # N x L * L x P -> N x P
