# Copyright 2018-2020 The GPflow Contributors. All Rights Reserved.
#
# 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.
r"""
Kernels form a core component of GPflow models and allow prior information to
be encoded about a latent function of interest. The effect of choosing
different kernels, and how it is possible to combine multiple kernels is shown
in the `"Using kernels in GPflow" notebook <notebooks/kernels.html>`_.
Broadcasting over leading dimensions:
`kernel.K(X1, X2)` returns the kernel evaluated on every pair in X1 and X2.
E.g. if X1 has shape [S1, N1, D] and X2 has shape [S2, N2, D], kernel.K(X1, X2)
will return a tensor of shape [S1, N1, S2, N2]. Similarly, kernel.K(X1, X1)
returns a tensor of shape [S1, N1, S1, N1]. In contrast, the return shape of
kernel.K(X1) is [S1, N1, N1]. (Without leading dimensions, the behaviour of
kernel.K(X, None) is identical to kernel.K(X, X).)
"""
import abc
from functools import partial, reduce
from typing import Callable, List, Optional, Sequence, Tuple, Union
import numpy as np
import tensorflow as tf
from ..base import Module, TensorType
ActiveDims = Union[slice, Sequence[int]]
NormalizedActiveDims = Union[slice, np.ndarray]
[docs]class Kernel(Module, metaclass=abc.ABCMeta):
"""
The basic kernel class. Handles active dims.
"""
def __init__(
self, active_dims: Optional[ActiveDims] = None, name: Optional[str] = None
) -> None:
"""
:param active_dims: active dimensions, either a slice or list of
indices into the columns of X.
:param name: optional kernel name.
"""
super().__init__(name=name)
self._active_dims = self._normalize_active_dims(active_dims)
@staticmethod
def _normalize_active_dims(value: Optional[ActiveDims]) -> NormalizedActiveDims:
if value is None:
value = slice(None, None, None)
if not isinstance(value, slice):
value = np.array(value, dtype=int)
return value
@property
def active_dims(self) -> NormalizedActiveDims:
return self._active_dims
@active_dims.setter
def active_dims(self, value: ActiveDims) -> None:
self._active_dims = self._normalize_active_dims(value)
[docs] def on_separate_dims(self, other: "Kernel") -> bool:
"""
Checks if the dimensions, over which the kernels are specified, overlap.
Returns True if they are defined on different/separate dimensions and False otherwise.
"""
if isinstance(self.active_dims, slice) or isinstance(other.active_dims, slice):
# Be very conservative for kernels defined over slices of dimensions
return False
if self.active_dims is None or other.active_dims is None:
return False
this_dims = self.active_dims.reshape(-1, 1)
other_dims = other.active_dims.reshape(1, -1)
return not np.any(this_dims == other_dims)
[docs] def slice(self, X: TensorType, X2: Optional[TensorType] = None) -> Tuple[tf.Tensor, tf.Tensor]:
"""
Slice the correct dimensions for use in the kernel, as indicated by `self.active_dims`.
:param X: Input 1 [N, D].
:param X2: Input 2 [M, D], can be None.
:return: Sliced X, X2, [N, I], I - input dimension.
"""
dims = self.active_dims
if isinstance(dims, slice):
X = X[..., dims]
if X2 is not None:
X2 = X2[..., dims]
elif dims is not None:
X = tf.gather(X, dims, axis=-1)
if X2 is not None:
X2 = tf.gather(X2, dims, axis=-1)
return X, X2
[docs] def slice_cov(self, cov: TensorType) -> tf.Tensor:
"""
Slice the correct dimensions for use in the kernel, as indicated by
`self.active_dims` for covariance matrices. This requires slicing the
rows *and* columns. This will also turn flattened diagonal
matrices into a tensor of full diagonal matrices.
:param cov: Tensor of covariance matrices, [N, D, D] or [N, D].
:return: [N, I, I].
"""
if cov.shape.ndims == 2:
cov = tf.linalg.diag(cov)
dims = self.active_dims
if isinstance(dims, slice):
return cov[..., dims, dims]
elif dims is not None:
nlast = tf.shape(cov)[-1]
ndims = len(dims)
cov_shape = tf.shape(cov)
cov_reshaped = tf.reshape(cov, [-1, nlast, nlast])
gather1 = tf.gather(tf.transpose(cov_reshaped, [2, 1, 0]), dims)
gather2 = tf.gather(tf.transpose(gather1, [1, 0, 2]), dims)
cov = tf.reshape(
tf.transpose(gather2, [2, 0, 1]), tf.concat([cov_shape[:-2], [ndims, ndims]], 0)
)
return cov
def _validate_ard_active_dims(self, ard_parameter: TensorType) -> None:
"""
Validate that ARD parameter matches the number of active_dims (provided active_dims
has been specified as an array).
"""
if self.active_dims is None or isinstance(self.active_dims, slice):
# Can only validate parameter if active_dims is an array
return
if ard_parameter.shape.rank > 0 and ard_parameter.shape[0] != len(self.active_dims):
raise ValueError(
f"Size of `active_dims` {self.active_dims} does not match "
f"size of ard parameter ({ard_parameter.shape[0]})"
)
@abc.abstractmethod
def K(self, X: TensorType, X2: Optional[TensorType] = None) -> tf.Tensor:
raise NotImplementedError
@abc.abstractmethod
def K_diag(self, X: TensorType) -> tf.Tensor:
raise NotImplementedError
def __call__(
self,
X: TensorType,
X2: Optional[TensorType] = None,
*,
full_cov: bool = True,
presliced: bool = False,
) -> tf.Tensor:
if (not full_cov) and (X2 is not None):
raise ValueError("Ambiguous inputs: `not full_cov` and `X2` are not compatible.")
if not presliced:
X, X2 = self.slice(X, X2)
if not full_cov:
assert X2 is None
return self.K_diag(X)
else:
return self.K(X, X2)
def __add__(self, other: "Kernel") -> "Kernel":
return Sum([self, other])
def __mul__(self, other: "Kernel") -> "Kernel":
return Product([self, other])
[docs]class Combination(Kernel):
"""
Combine a list of kernels, e.g. by adding or multiplying (see inheriting
classes).
The names of the kernels to be combined are generated from their class
names.
"""
_reduction = None
def __init__(self, kernels: Sequence[Kernel], name: Optional[str] = None) -> None:
super().__init__(name=name)
if not all(isinstance(k, Kernel) for k in kernels):
raise TypeError("can only combine Kernel instances") # pragma: no cover
self.kernels: List[Kernel] = []
self._set_kernels(kernels)
def _set_kernels(self, kernels: Sequence[Kernel]) -> None:
# add kernels to a list, flattening out instances of this class therein
kernels_list: List[Kernel] = []
for k in kernels:
if isinstance(k, self.__class__):
kernels_list.extend(k.kernels)
else:
kernels_list.append(k)
self.kernels = kernels_list
@property
def on_separate_dimensions(self) -> bool:
"""
Checks whether the kernels in the combination act on disjoint subsets
of dimensions. Currently, it is hard to asses whether two slice objects
will overlap, so this will always return False.
:return: Boolean indicator.
"""
if np.any([isinstance(k.active_dims, slice) for k in self.kernels]):
# Be conservative in the case of a slice object
return False
else:
dimlist = [k.active_dims for k in self.kernels]
overlapping = False
for i, dims_i in enumerate(dimlist):
assert isinstance(dims_i, np.ndarray)
for dims_j in dimlist[i + 1 :]:
assert isinstance(dims_j, np.ndarray)
if np.any(dims_i.reshape(-1, 1) == dims_j.reshape(1, -1)):
overlapping = True
return not overlapping
[docs]class ReducingCombination(Combination):
def __call__(
self,
X: TensorType,
X2: Optional[TensorType] = None,
*,
full_cov: bool = True,
presliced: bool = False,
) -> tf.Tensor:
return self._reduce(
[k(X, X2, full_cov=full_cov, presliced=presliced) for k in self.kernels]
)
def K(self, X: TensorType, X2: Optional[TensorType] = None) -> tf.Tensor:
return self._reduce([k.K(X, X2) for k in self.kernels])
def K_diag(self, X: TensorType) -> tf.Tensor:
return self._reduce([k.K_diag(X) for k in self.kernels])
@property
@abc.abstractmethod
def _reduce(self) -> Callable[[Sequence[TensorType]], TensorType]:
pass
[docs]class Sum(ReducingCombination):
@property
def _reduce(self) -> Callable[[Sequence[TensorType]], TensorType]:
return tf.add_n
[docs]class Product(ReducingCombination):
@property
def _reduce(self) -> Callable[[Sequence[TensorType]], TensorType]:
return partial(reduce, tf.multiply)