Revision 291ae6c7dbfcbded27c604f136982a5067d14b8e authored by thevincentadam on 20 January 2020, 12:17:20 UTC, committed by thevincentadam on 20 January 2020, 12:17:20 UTC
1 parent 5dc31b8
squared_exponentials.py
``````import tensorflow as tf

from .. import kernels
from .. import mean_functions as mfn
from ..inducing_variables import InducingPoints
from ..probability_distributions import DiagonalGaussian, Gaussian, MarkovGaussian
from ..utilities.ops import square_distance
from . import dispatch
from .expectations import expectation

NoneType = type(None)

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

:return: N
"""
return kernel(p.mu, full=False)

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

:return: NxM
"""
# use only active dimensions
Xcov = kernel.slice_cov(p.cov)
Z, Xmu = kernel.slice(inducing_variable.Z, p.mu)
D = tf.shape(Xmu)[1]

lengthscale = kernel.lengthscale
if not kernel.ard:
lengthscale = tf.zeros((D, ), dtype=lengthscale.dtype) + kernel.lengthscale

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

all_diffs = tf.transpose(Z) - 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(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

return kernel.variance * (determinants[:, None] * exponent_mahalanobis)

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

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

D = tf.shape(Xmu)[1]

lengthscale = kernel.lengthscale
if not kernel.ard:
lengthscale = tf.zeros((D, ), dtype=lengthscale.dtype) + lengthscale

chol_L_plus_Xcov = tf.linalg.cholesky(tf.linalg.diag(lengthscale**2) + Xcov)  # NxDxD
all_diffs = tf.transpose(inducing_variable.Z) - tf.expand_dims(Xmu, 2)  # NxDxM

sqrt_det_L = tf.reduce_prod(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

exponent_mahalanobis = tf.linalg.cholesky_solve(chol_L_plus_Xcov, all_diffs)  # NxDxM
non_exponent_term = tf.linalg.matmul(Xcov, exponent_mahalanobis, transpose_a=True)
non_exponent_term = tf.expand_dims(Xmu, 2) + non_exponent_term  # NxDxM

exponent_mahalanobis = tf.reduce_sum(all_diffs * exponent_mahalanobis, 1)  # NxM
exponent_mahalanobis = tf.exp(-0.5 * exponent_mahalanobis)  # NxM

return kernel.variance * (determinants[:, None] * exponent_mahalanobis)[:, None, :] * non_exponent_term

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

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

D = tf.shape(Xmu)[1]
lengthscale = kernel.lengthscale
if not kernel.ard:
lengthscale = tf.zeros((D, ), dtype=lengthscale.dtype) + lengthscale

chol_L_plus_Xcov = tf.linalg.cholesky(tf.linalg.diag(lengthscale**2) + Xcov[0, :-1])  # NxDxD
all_diffs = tf.transpose(inducing_variable.Z) - tf.expand_dims(Xmu[:-1], 2)  # NxDxM

sqrt_det_L = tf.reduce_prod(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

exponent_mahalanobis = tf.linalg.cholesky_solve(chol_L_plus_Xcov, all_diffs)  # NxDxM
non_exponent_term = tf.linalg.matmul(Xcov[1, :-1], exponent_mahalanobis, transpose_a=True)
non_exponent_term = tf.expand_dims(Xmu[1:], 2) + non_exponent_term  # NxDxM

exponent_mahalanobis = tf.reduce_sum(all_diffs * exponent_mahalanobis, 1)  # NxM
exponent_mahalanobis = tf.exp(-0.5 * exponent_mahalanobis)  # NxM

return kernel.variance * (determinants[:, None] * exponent_mahalanobis)[:, None, :] * non_exponent_term

@dispatch.expectation.register((Gaussian, DiagonalGaussian), kernels.SquaredExponential, InducingPoints, kernels.SquaredExponential, 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_{.,.} :: RBF 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 feat1 != feat2 or kern1 != kern2:
raise NotImplementedError("The expectation over two kernels has only an "
"analytical implementation if both kernels are equal.")

kernel = kern1
inducing_variable = 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(inducing_variable.Z, p.mu)

N = tf.shape(Xmu)[0]
D = tf.shape(Xmu)[1]

squared_lengthscale = kernel.lengthscale**2
if not kernel.ard:
zero_lengthscale = tf.zeros((D, ), dtype=squared_lengthscale.dtype)
squared_lengthscale = squared_lengthscale + zero_lengthscale

sqrt_det_L = tf.reduce_prod(0.5 * squared_lengthscale)**0.5
C = tf.linalg.cholesky(0.5 * tf.linalg.diag(squared_lengthscale) + Xcov)  # NxDxD
dets = sqrt_det_L / tf.exp(tf.reduce_sum(tf.math.log(tf.linalg.diag_part(C)), axis=1))  # N

C_inv_mu = tf.linalg.triangular_solve(C, tf.expand_dims(Xmu, 2), lower=True)  # NxDx1
C_inv_z = tf.linalg.triangular_solve(C, tf.tile(tf.expand_dims(0.5 * tf.transpose(Z), 0), [N, 1, 1]),
lower=True)  # NxDxM
mu_CC_inv_mu = tf.expand_dims(tf.reduce_sum(tf.square(C_inv_mu), 1), 2)  # Nx1x1
z_CC_inv_z = tf.reduce_sum(tf.square(C_inv_z), 1)  # NxM
zm_CC_inv_zn = tf.linalg.matmul(C_inv_z, C_inv_z, transpose_a=True)  # NxMxM
two_z_CC_inv_mu = 2 * tf.linalg.matmul(C_inv_z, C_inv_mu, transpose_a=True)[:, :, 0]  # NxM
# NxMxM
exponent_mahalanobis = mu_CC_inv_mu + tf.expand_dims(z_CC_inv_z, 1) + \
tf.expand_dims(z_CC_inv_z, 2) + 2 * zm_CC_inv_zn - \
tf.expand_dims(two_z_CC_inv_mu, 2) - tf.expand_dims(two_z_CC_inv_mu, 1)
exponent_mahalanobis = tf.exp(-0.5 * exponent_mahalanobis)  # NxMxM

# Compute sqrt(self(Z)) explicitly to prevent automatic gradient from
# being NaN sometimes, see pull request #615
kernel_sqrt = tf.exp(-0.25 * square_distance(Z / kernel.lengthscale, None))
return kernel.variance**2 * kernel_sqrt * tf.reshape(dets, [N, 1, 1]) * exponent_mahalanobis
``````

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