Source code for gpflow.kernels.multioutput.kernels

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import abc
from typing import Optional, Sequence, Tuple

import tensorflow as tf

from ...base import Parameter, TensorType
from ..base import Combination, Kernel


[docs]class MultioutputKernel(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, P, N, P] if full_output_cov = True - [P, N, N] if full_output_cov = False and `K_diag` returns: - [N, P, P] if full_output_cov = True - [N, 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. """ @property @abc.abstractmethod def num_latent_gps(self) -> int: """The number of latent GPs in the multioutput kernel""" raise NotImplementedError @property @abc.abstractmethod def latent_kernels(self) -> Tuple[Kernel, ...]: """The underlying kernels in the multioutput kernel""" raise NotImplementedError
[docs] @abc.abstractmethod def K( self, X: TensorType, X2: Optional[TensorType] = None, full_output_cov: bool = True ) -> tf.Tensor: """ Returns the correlation of f(X) and f(X2), where f(.) can be multi-dimensional. :param X: data matrix, [N1, D] :param X2: data matrix, [N2, D] :param full_output_cov: calculate correlation between outputs. :return: cov[f(X), f(X2)] with shape - [N1, P, N2, P] if `full_output_cov` = True - [P, N1, N2] if `full_output_cov` = False """ raise NotImplementedError
[docs] @abc.abstractmethod def K_diag(self, X: TensorType, full_output_cov: bool = True) -> tf.Tensor: """ Returns the correlation of f(X) and f(X), where f(.) can be multi-dimensional. :param X: data matrix, [N, D] :param full_output_cov: calculate correlation between outputs. :return: var[f(X)] with shape - [N, P, N, P] if `full_output_cov` = True - [N, P] if `full_output_cov` = False """ raise NotImplementedError
def __call__( self, X: TensorType, X2: Optional[TensorType] = None, *, full_cov: bool = False, full_output_cov: bool = True, presliced: bool = False, ) -> tf.Tensor: if not presliced: X, X2 = self.slice(X, X2) if not full_cov and X2 is not None: raise ValueError( "Ambiguous inputs: passing in `X2` is not compatible with `full_cov=False`." ) if not full_cov: return self.K_diag(X, full_output_cov=full_output_cov) return self.K(X, X2, full_output_cov=full_output_cov)
[docs]class SharedIndependent(MultioutputKernel): """ - 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, kernel: Kernel, output_dim: int) -> None: super().__init__() self.kernel = kernel self.output_dim = output_dim @property def num_latent_gps(self) -> int: # In this case number of latent GPs (L) == output_dim (P) return self.output_dim @property def latent_kernels(self) -> Tuple[Kernel, ...]: """The underlying kernels in the multioutput kernel""" return (self.kernel,)
[docs] def K( self, X: TensorType, X2: Optional[TensorType] = None, full_output_cov: bool = True ) -> tf.Tensor: K = self.kernel.K(X, X2) # [N, N2] if full_output_cov: Ks = tf.tile(K[..., None], [1, 1, self.output_dim]) # [N, N2, P] return tf.transpose(tf.linalg.diag(Ks), [0, 2, 1, 3]) # [N, P, N2, P] else: return tf.tile(K[None, ...], [self.output_dim, 1, 1]) # [P, N, N2]
[docs] def K_diag(self, X: TensorType, full_output_cov: bool = True) -> tf.Tensor: K = self.kernel.K_diag(X) # N Ks = tf.tile(K[:, None], [1, self.output_dim]) # [N, P] return tf.linalg.diag(Ks) if full_output_cov else Ks # [N, P, P] or [N, P]
[docs]class SeparateIndependent(MultioutputKernel, Combination): """ - Separate: we use different kernel for each output latent - Independent: Latents are uncorrelated a priori. """ def __init__(self, kernels: Sequence[Kernel], name: Optional[str] = None) -> None: super().__init__(kernels=kernels, name=name) @property def num_latent_gps(self) -> int: return len(self.kernels) @property def latent_kernels(self) -> Tuple[Kernel, ...]: """The underlying kernels in the multioutput kernel""" return tuple(self.kernels)
[docs] def K( self, X: TensorType, X2: Optional[TensorType] = None, full_output_cov: bool = True ) -> tf.Tensor: if full_output_cov: Kxxs = tf.stack([k.K(X, X2) for k in self.kernels], axis=2) # [N, N2, P] return tf.transpose(tf.linalg.diag(Kxxs), [0, 2, 1, 3]) # [N, P, N2, P] else: return tf.stack([k.K(X, X2) for k in self.kernels], axis=0) # [P, N, N2]
[docs] def K_diag(self, X: TensorType, full_output_cov: bool = False) -> tf.Tensor: stacked = tf.stack([k.K_diag(X) for k in self.kernels], axis=1) # [N, P] return tf.linalg.diag(stacked) if full_output_cov else stacked # [N, P, P] or [N, P]
[docs]class IndependentLatent(MultioutputKernel): """ Base class for multioutput kernels that are constructed from independent latent Gaussian processes. It should always be possible to specify inducing variables for such kernels that give a block-diagonal Kuu, which can be represented as a [L, M, M] tensor. A reasonable (but not optimal) inference procedure can be specified by placing the inducing points in the latent processes and simply computing Kuu [L, M, M] and Kuf [N, P, M, L] and using `fallback_independent_latent_ conditional()`. This can be specified by using `Fallback{Separate|Shared} IndependentInducingVariables`. """ @abc.abstractmethod def Kgg(self, X: TensorType, X2: TensorType) -> tf.Tensor: raise NotImplementedError
[docs]class LinearCoregionalization(IndependentLatent, Combination): """ Linear mixing of the latent GPs to form the output. """ def __init__(self, kernels: Sequence[Kernel], W: TensorType, name: Optional[str] = None): Combination.__init__(self, kernels=kernels, name=name) self.W = Parameter(W) # [P, L] @property def num_latent_gps(self) -> int: return self.W.shape[-1] # L @property def latent_kernels(self) -> Tuple[Kernel, ...]: """The underlying kernels in the multioutput kernel""" return tuple(self.kernels) def Kgg(self, X: TensorType, X2: TensorType) -> tf.Tensor: return tf.stack([k.K(X, X2) for k in self.kernels], axis=0) # [L, N, N2]
[docs] def K( self, X: TensorType, X2: Optional[TensorType] = None, full_output_cov: bool = True ) -> tf.Tensor: Kxx = self.Kgg(X, X2) # [L, N, N2] KxxW = Kxx[None, :, :, :] * self.W[:, :, None, None] # [P, L, N, 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, P, N, N2] return tf.transpose(WKxxW, [2, 0, 3, 1]) # [N, P, N2, 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, N, N2]
[docs] def K_diag(self, X: TensorType, full_output_cov: bool = True) -> tf.Tensor: K = tf.stack([k.K_diag(X) for k in self.kernels], axis=1) # [N, 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, P, P] Wt = tf.transpose(self.W) # [L, P] return tf.reduce_sum( K[:, :, None, None] * Wt[None, :, :, None] * Wt[None, :, None, :], axis=1 ) # [N, P, P] else: # return tf.einsum('nl,lk,lk->nkq', K, self.W, self.W) # [N, P] return tf.linalg.matmul( K, self.W ** 2.0, transpose_b=True ) # [N, L] * [L, P] -> [N, P]