linears.py
``````import tensorflow as tf

from .. import kernels
from .. import mean_functions as mfn
from ..features import InducingPoints
from ..probability_distributions import DiagonalGaussian, Gaussian, MarkovGaussian
from . import dispatch
from .expectations import expectation

NoneType = type(None)

@dispatch.expectation.register(Gaussian, kernels.Linear, NoneType, NoneType, NoneType)
def _E(p, kernel, _, __, ___, nghp=None):
"""
Compute the expectation:
<diag(K_{X, X})>_p(X)
- K_{.,.} :: Linear kernel

:return: N
"""
# use only active dimensions
Xmu, _ = kernel.slice(p.mu, None)
Xcov = kernel.slice_cov(p.cov)

return tf.reduce_sum(kernel.variance * (tf.linalg.diag_part(Xcov) + Xmu**2), 1)

@dispatch.expectation.register(Gaussian, kernels.Linear, InducingPoints, NoneType, NoneType)
def _E(p, kernel, feature, _, __, nghp=None):
"""
Compute the expectation:
<K_{X, Z}>_p(X)
- K_{.,.} :: Linear kernel

:return: NxM
"""
# use only active dimensions
Z, Xmu = kernel.slice(feature.Z, p.mu)

return tf.linalg.matmul(Xmu, Z * kernel.variance, transpose_b=True)

@dispatch.expectation.register(Gaussian, kernels.Linear, InducingPoints, mfn.Identity, NoneType)
def _E(p, kernel, feature, mean, _, nghp=None):
"""
Compute the expectation:
expectation[n] = <K_{Z, x_n} x_n^T>_p(x_n)
- K_{.,.} :: Linear kernel

:return: NxMxD
"""
Xmu, Xcov = p.mu, p.cov

N = Xmu.shape[0]
var_Z = kernel.variance * feature.Z  # MxD
tiled_Z = tf.tile(tf.expand_dims(var_Z, 0), (N, 1, 1))  # NxMxD
return tf.linalg.matmul(tiled_Z, Xcov + (Xmu[..., None] * Xmu[:, None, :]))

@dispatch.expectation.register(MarkovGaussian, kernels.Linear, InducingPoints, mfn.Identity, NoneType)
def _E(p, kernel, feature, mean, _, nghp=None):
"""
Compute the expectation:
expectation[n] = <K_{Z, x_n} x_{n+1}^T>_p(x_{n:n+1})
- K_{.,.} :: Linear kernel
- p       :: MarkovGaussian distribution (p.cov 2x(N+1)xDxD)

:return: NxMxD
"""
Xmu, Xcov = p.mu, p.cov

N = Xmu.shape[0] - 1
var_Z = kernel.variance * feature.Z  # MxD
tiled_Z = tf.tile(tf.expand_dims(var_Z, 0), (N, 1, 1))  # NxMxD
eXX = Xcov[1, :-1] + (Xmu[:-1][..., None] * Xmu[1:][:, None, :])  # NxDxD
return tf.linalg.matmul(tiled_Z, eXX)

@dispatch.expectation.register((Gaussian, DiagonalGaussian), kernels.Linear, InducingPoints, kernels.Linear,
InducingPoints)
def _E(p, kern1, feat1, kern2, feat2, nghp=None):
"""
Compute the expectation:
expectation[n] = <Ka_{Z1, x_n} Kb_{x_n, Z2}>_p(x_n)
- Ka_{.,.}, Kb_{.,.} :: Linear kernels
Ka and Kb as well as Z1 and Z2 can differ from each other, but this is supported
only if the Gaussian p is Diagonal (p.cov NxD) and Ka, Kb have disjoint active_dims
in which case the joint expectations simplify into a product of expectations

:return: NxMxM
"""
if kern1.on_separate_dims(kern2) and isinstance(p, DiagonalGaussian):  # no joint expectations required
eKxz1 = expectation(p, (kern1, feat1))
eKxz2 = expectation(p, (kern2, feat2))
return eKxz1[:, :, None] * eKxz2[:, None, :]

if kern1 != kern2 or feat1 != feat2:
raise NotImplementedError("The expectation over two kernels has only an "
"analytical implementation if both kernels are equal.")

kernel = kern1
feature = feat1

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

N = Xmu.shape[0]
var_Z = kernel.variance * Z
tiled_Z = tf.tile(tf.expand_dims(var_Z, 0), (N, 1, 1))  # NxMxD
XX = Xcov + tf.expand_dims(Xmu, 1) * tf.expand_dims(Xmu, 2)  # NxDxD
return tf.linalg.matmul(tf.linalg.matmul(tiled_Z, XX), tiled_Z, transpose_b=True)
``````