cross_kernels.py
``````import tensorflow as tf

from . import dispatch
from .. import kernels
from ..features import InducingPoints
from ..probability_distributions import DiagonalGaussian, Gaussian
from .expectations import expectation

@dispatch.expectation.register((Gaussian, DiagonalGaussian), kernels.RBF,
InducingPoints, kernels.Linear, InducingPoints)
def _E(p, rbf_kern, feat1, lin_kern, feat2, nghp=None):
"""
Compute the expectation:
expectation[n] = <Ka_{Z1, x_n} Kb_{x_n, Z2}>_p(x_n)
- K_lin_{.,.} :: RBF kernel
- K_rbf_{.,.} :: Linear kernel
Different Z1 and Z2 are handled if p is diagonal and K_lin and K_rbf have disjoint
active_dims, in which case the joint expectations simplify into a product of expectations

:return: NxM1xM2
"""
if rbf_kern.on_separate_dims(lin_kern) and isinstance(
p, DiagonalGaussian):  # no joint expectations required
eKxz1 = expectation(p, (rbf_kern, feat1))
eKxz2 = expectation(p, (lin_kern, feat2))
return eKxz1[:, :, None] * eKxz2[:, None, :]

if feat1 != feat2:
raise NotImplementedError(
"Features have to be the same for both kernels.")

if rbf_kern.active_dims != lin_kern.active_dims:
raise NotImplementedError(
"active_dims have to be the same for both kernels.")

# use only active dimensions
Xcov = rbf_kern.slice_cov(
tf.linalg.diag(p.cov) if isinstance(p, DiagonalGaussian) else p.cov)
Z, Xmu = rbf_kern.slice(feat1.Z, p.mu)

N = Xmu.shape[0]
D = Xmu.shape[1]

def take_with_ard(value):
if not rbf_kern.ard:
return tf.zeros((D, ), dtype=value.dtype) + value
return value

lin_kern_variances = take_with_ard(lin_kern.variance)
rbf_kern_lengthscale = take_with_ard(rbf_kern.lengthscale)

chol_L_plus_Xcov = tf.linalg.cholesky(
tf.linalg.diag(rbf_kern_lengthscale**2) + Xcov)  # NxDxD

Z_transpose = tf.transpose(Z)
all_diffs = Z_transpose - tf.expand_dims(Xmu, 2)  # NxDxM
exponent_mahalanobis = tf.linalg.triangular_solve(chol_L_plus_Xcov,
all_diffs,
lower=True)  # NxDxM
exponent_mahalanobis = tf.reduce_sum(tf.square(exponent_mahalanobis),
1)  # NxM
exponent_mahalanobis = tf.exp(-0.5 * exponent_mahalanobis)  # NxM

sqrt_det_L = tf.reduce_prod(rbf_kern_lengthscale)
sqrt_det_L_plus_Xcov = tf.exp(
tf.reduce_sum(tf.math.log(tf.linalg.diag_part(chol_L_plus_Xcov)),
axis=1))
determinants = sqrt_det_L / sqrt_det_L_plus_Xcov  # N
eKxz_rbf = rbf_kern.variance * (
determinants[:, None] * exponent_mahalanobis
)  ## NxM <- End RBF eKxz code

tiled_Z = tf.tile(tf.expand_dims(Z_transpose, 0), (N, 1, 1))  # NxDxM
z_L_inv_Xcov = tf.linalg.matmul(tiled_Z,
Xcov / rbf_kern_lengthscale[:, None]**2.,
transpose_a=True)  # NxMxD

cross_eKzxKxz = tf.linalg.cholesky_solve(
chol_L_plus_Xcov,
(lin_kern_variances * rbf_kern_lengthscale**2.)[..., None] *
tiled_Z)  # NxDxM

cross_eKzxKxz = tf.linalg.matmul(
(z_L_inv_Xcov + Xmu[:, None, :]) * eKxz_rbf[..., None],
cross_eKzxKxz)  # NxMxM
return cross_eKzxKxz

@dispatch.expectation.register((Gaussian, DiagonalGaussian), kernels.Linear,
InducingPoints, kernels.RBF, InducingPoints)
def _E(p, lin_kern, feat1, rbf_kern, feat2, nghp=None):
"""
Compute the expectation:
expectation[n] = <Ka_{Z1, x_n} Kb_{x_n, Z2}>_p(x_n)
- K_lin_{.,.} :: Linear kernel
- K_rbf_{.,.} :: RBF kernel
Different Z1 and Z2 are handled if p is diagonal and K_lin and K_rbf have disjoint
active_dims, in which case the joint expectations simplify into a product of expectations

:return: NxM1xM2
"""