# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
# Licensed under the BSD 3-clause license (see LICENSE.txt)
"""
The package for the Psi statistics computation of the linear kernel for SSGPLVM
"""
from ....util.linalg import tdot
import numpy as np
[docs]def psicomputations(variance, Z, variational_posterior, return_psi2_n=False):
"""
Compute psi-statistics for ss-linear kernel
"""
# here are the "statistics" for psi0, psi1 and psi2
# Produced intermediate results:
# psi0 N
# psi1 NxM
# psi2 MxM
mu = variational_posterior.mean
S = variational_posterior.variance
gamma = variational_posterior.binary_prob
psi0 = (gamma*(np.square(mu)+S)*variance).sum(axis=-1)
psi1 = np.inner(variance*gamma*mu,Z)
psi2 = np.inner(np.square(variance)*(gamma*((1-gamma)*np.square(mu)+S)).sum(axis=0)*Z,Z)+tdot(psi1.T)
return psi0, psi1, psi2
[docs]def psiDerivativecomputations(dL_dpsi0, dL_dpsi1, dL_dpsi2, variance, Z, variational_posterior):
mu = variational_posterior.mean
S = variational_posterior.variance
gamma = variational_posterior.binary_prob
dL_dvar, dL_dgamma, dL_dmu, dL_dS, dL_dZ = _psi2computations(dL_dpsi2, variance, Z, mu, S, gamma)
# Compute for psi0 and psi1
mu2S = np.square(mu)+S
dL_dvar += (dL_dpsi0[:,None]*gamma*mu2S).sum(axis=0) + (dL_dpsi1.T.dot(gamma*mu)*Z).sum(axis=0)
dL_dgamma += dL_dpsi0[:,None]*variance*mu2S+ dL_dpsi1.dot(Z)*mu*variance
dL_dmu += dL_dpsi0[:,None]*2.*variance*gamma*mu + dL_dpsi1.dot(Z)*gamma*variance
dL_dS += dL_dpsi0[:,None]*variance*gamma
dL_dZ += dL_dpsi1.T.dot(gamma*mu)*variance
return dL_dvar, dL_dZ, dL_dmu, dL_dS, dL_dgamma
def _psi2computations(dL_dpsi2, variance, Z, mu, S, gamma):
"""
Z - MxQ
mu - NxQ
S - NxQ
gamma - NxQ
"""
# here are the "statistics" for psi1 and psi2
# Produced intermediate results:
# _psi2_dvariance Q
# _psi2_dZ MxQ
# _psi2_dgamma NxQ
# _psi2_dmu NxQ
# _psi2_dS NxQ
mu2 = np.square(mu)
gamma2 = np.square(gamma)
variance2 = np.square(variance)
mu2S = mu2+S # NxQ
gvm = gamma*mu*variance
common_sum = gvm.dot(Z.T)
Z_expect = (np.dot(dL_dpsi2,Z)*Z).sum(axis=0)
Z_expect_var2 = Z_expect*variance2
dL_dpsi2T = dL_dpsi2+dL_dpsi2.T
common_expect = common_sum.dot(dL_dpsi2T).dot(Z)
Z2_expect = common_sum.dot(dL_dpsi2T)
Z1_expect = dL_dpsi2T.dot(Z)
dL_dvar = variance*Z_expect*2.*(gamma*mu2S-gamma2*mu2).sum(axis=0)+(common_expect*gamma*mu).sum(axis=0)
dL_dgamma = Z_expect_var2*(mu2S-2.*gamma*mu2)+common_expect*mu*variance
dL_dmu = Z_expect_var2*mu*2.*(gamma-gamma2) + common_expect*gamma*variance
dL_dS = gamma*Z_expect_var2
dL_dZ = (gamma*(mu2S-gamma*mu2)).sum(axis=0)*variance2*Z1_expect+ Z2_expect.T.dot(gamma*mu)*variance
return dL_dvar, dL_dgamma, dL_dmu, dL_dS, dL_dZ