# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
# Licensed under the BSD 3-clause license (see LICENSE.txt)
from .posterior import Posterior
from ...util.linalg import mdot, jitchol, backsub_both_sides, tdot, dtrtrs, dtrtri, dpotri, dpotrs, symmetrify
from ...util import diag
from GPy.core.parameterization.variational import VariationalPosterior
import numpy as np
from . import LatentFunctionInference
log_2_pi = np.log(2*np.pi)
import logging, itertools
logger = logging.getLogger('vardtc')
[docs]class VarDTC(LatentFunctionInference):
"""
An object for inference when the likelihood is Gaussian, but we want to do sparse inference.
The function self.inference returns a Posterior object, which summarizes
the posterior.
For efficiency, we sometimes work with the cholesky of Y*Y.T. To save repeatedly recomputing this, we cache it.
"""
const_jitter = 1e-8
def __init__(self, limit=1):
from paramz.caching import Cacher
self.limit = limit
self.get_trYYT = Cacher(self._get_trYYT, limit)
self.get_YYTfactor = Cacher(self._get_YYTfactor, limit)
[docs] def set_limit(self, limit):
self.get_trYYT.limit = limit
self.get_YYTfactor.limit = limit
def _get_trYYT(self, Y):
return np.einsum("ij,ij->", Y, Y)
# faster than, but same as:
# return np.sum(np.square(Y))
def __getstate__(self):
# has to be overridden, as Cacher objects cannot be pickled.
return self.limit
def __setstate__(self, state):
# has to be overridden, as Cacher objects cannot be pickled.
self.limit = state
from paramz.caching import Cacher
self.get_trYYT = Cacher(self._get_trYYT, self.limit)
self.get_YYTfactor = Cacher(self._get_YYTfactor, self.limit)
def _get_YYTfactor(self, Y):
"""
find a matrix L which satisfies LLT = YYT.
Note that L may have fewer columns than Y.
"""
N, D = Y.shape
if (N>=D):
return Y.view(np.ndarray)
else:
return jitchol(tdot(Y))
[docs] def get_VVTfactor(self, Y, prec):
return Y * prec # TODO cache this, and make it effective
[docs] def inference(self, kern, X, Z, likelihood, Y, Y_metadata=None, mean_function=None, precision=None, Lm=None, dL_dKmm=None, psi0=None, psi1=None, psi2=None, Z_tilde=None):
num_data, output_dim = Y.shape
num_inducing = Z.shape[0]
uncertain_inputs = isinstance(X, VariationalPosterior)
if mean_function is not None:
mean = mean_function.f(X)
else:
mean = 0
if precision is None:
#assume Gaussian likelihood
precision = 1./np.fmax(likelihood.gaussian_variance(Y_metadata), self.const_jitter)
if precision.ndim == 1:
precision = precision[:, None]
het_noise = precision.size > 1
if (het_noise or uncertain_inputs) and mean_function is not None:
raise ValueError('Mean function not implemented with uncertain inputs or heteroscedasticity')
VVT_factor = precision*(Y-mean)
trYYT = self.get_trYYT(Y-mean)
# kernel computations, using BGPLVM notation
if Lm is None:
Kmm = kern.K(Z).copy()
diag.add(Kmm, self.const_jitter)
Lm = jitchol(Kmm)
else:
Kmm = tdot(Lm)
symmetrify(Kmm)
# The rather complex computations of A, and the psi stats
if uncertain_inputs:
if psi0 is None:
psi0 = kern.psi0(Z, X)
if psi1 is None:
psi1 = kern.psi1(Z, X)
if het_noise:
if psi2 is None:
psi2_beta = (kern.psi2n(Z, X) * precision[:, :, None]).sum(0)
else:
psi2_beta = (psi2 * precision[:, :, None]).sum(0)
else:
if psi2 is None:
psi2_beta = kern.psi2(Z,X) * precision
elif psi2.ndim == 3:
psi2_beta = psi2.sum(0) * precision
else:
psi2_beta = psi2 * precision
LmInv = dtrtri(Lm)
A = LmInv.dot(psi2_beta.dot(LmInv.T))
else:
if psi0 is None:
psi0 = kern.Kdiag(X)
if psi1 is None:
psi1 = kern.K(X, Z)
if het_noise:
tmp = psi1 * (np.sqrt(precision))
else:
tmp = psi1 * (np.sqrt(precision))
tmp, _ = dtrtrs(Lm, tmp.T, lower=1)
A = tdot(tmp) #print A.sum()
# factor B
B = np.eye(num_inducing) + A
LB = jitchol(B)
# back substutue C into psi1Vf
tmp, _ = dtrtrs(Lm, psi1.T, lower=1, trans=0)
_LBi_Lmi_psi1, _ = dtrtrs(LB, tmp, lower=1, trans=0)
_LBi_Lmi_psi1Vf = np.dot(_LBi_Lmi_psi1, VVT_factor)
tmp, _ = dtrtrs(LB, _LBi_Lmi_psi1Vf, lower=1, trans=1)
Cpsi1Vf, _ = dtrtrs(Lm, tmp, lower=1, trans=1)
# data fit and derivative of L w.r.t. Kmm
dL_dm = -_LBi_Lmi_psi1.T.dot(_LBi_Lmi_psi1.dot(VVT_factor)) + VVT_factor
delit = tdot(_LBi_Lmi_psi1Vf)
data_fit = np.trace(delit)
DBi_plus_BiPBi = backsub_both_sides(LB, output_dim * np.eye(num_inducing) + delit)
if dL_dKmm is None:
delit = -0.5 * DBi_plus_BiPBi
delit += -0.5 * B * output_dim
delit += output_dim * np.eye(num_inducing)
# Compute dL_dKmm
dL_dKmm = backsub_both_sides(Lm, delit)
# derivatives of L w.r.t. psi
dL_dpsi0, dL_dpsi1, dL_dpsi2 = _compute_dL_dpsi(num_inducing, num_data, output_dim, precision, Lm,
VVT_factor, Cpsi1Vf, DBi_plus_BiPBi,
psi1, het_noise, uncertain_inputs)
# log marginal likelihood
log_marginal = _compute_log_marginal_likelihood(likelihood, num_data, output_dim, precision, het_noise,
psi0, A, LB, trYYT, data_fit, Y)
if Z_tilde is not None:
# This is a correction term for the log marginal likelihood
# In EP this is log Z_tilde, which is the difference between the
# Gaussian marginal and Z_EP
log_marginal += Z_tilde
#noise derivatives
dL_dR = _compute_dL_dR(likelihood,
het_noise, uncertain_inputs, LB,
_LBi_Lmi_psi1Vf, DBi_plus_BiPBi, Lm, A,
psi0, psi1, precision,
data_fit, num_data, output_dim, trYYT, Y, VVT_factor)
dL_dthetaL = likelihood.exact_inference_gradients(dL_dR,Y_metadata)
#put the gradients in the right places
if uncertain_inputs:
grad_dict = {'dL_dKmm': dL_dKmm,
'dL_dpsi0':dL_dpsi0,
'dL_dpsi1':dL_dpsi1,
'dL_dpsi2':dL_dpsi2,
'dL_dthetaL':dL_dthetaL}
else:
grad_dict = {'dL_dKmm': dL_dKmm,
'dL_dKdiag':dL_dpsi0,
'dL_dKnm':dL_dpsi1,
'dL_dthetaL':dL_dthetaL,
'dL_dm':dL_dm}
#get sufficient things for posterior prediction
#TODO: do we really want to do this in the loop?
if VVT_factor.shape[1] == Y.shape[1]:
woodbury_vector = Cpsi1Vf # == Cpsi1V
else:
print('foobar')
import ipdb; ipdb.set_trace()
psi1V = np.dot(Y.T*precision, psi1).T
tmp, _ = dtrtrs(Lm, psi1V, lower=1, trans=0)
tmp, _ = dpotrs(LB, tmp, lower=1)
woodbury_vector, _ = dtrtrs(Lm, tmp, lower=1, trans=1)
#Bi, _ = dpotri(LB, lower=1)
#symmetrify(Bi)
Bi = -dpotri(LB, lower=1)[0]
diag.add(Bi, 1)
woodbury_inv = backsub_both_sides(Lm, Bi)
#construct a posterior object
post = Posterior(woodbury_inv=woodbury_inv, woodbury_vector=woodbury_vector, K=Kmm, mean=None, cov=None, K_chol=Lm)
return post, log_marginal, grad_dict
def _compute_dL_dpsi(num_inducing, num_data, output_dim, beta, Lm, VVT_factor, Cpsi1Vf, DBi_plus_BiPBi, psi1, het_noise, uncertain_inputs):
dL_dpsi0 = -0.5 * output_dim * (beta* np.ones([num_data, 1])).flatten()
dL_dpsi1 = np.dot(VVT_factor, Cpsi1Vf.T)
dL_dpsi2_beta = 0.5 * backsub_both_sides(Lm, output_dim * np.eye(num_inducing) - DBi_plus_BiPBi)
if het_noise:
if uncertain_inputs:
dL_dpsi2 = beta[:, None] * dL_dpsi2_beta[None, :, :]
else:
dL_dpsi1 += 2.*np.dot(dL_dpsi2_beta, (psi1 * beta).T).T
dL_dpsi2 = None
else:
dL_dpsi2 = beta * dL_dpsi2_beta
if not uncertain_inputs:
# subsume back into psi1 (==Kmn)
dL_dpsi1 += 2.*np.dot(psi1, dL_dpsi2)
dL_dpsi2 = None
return dL_dpsi0, dL_dpsi1, dL_dpsi2
def _compute_dL_dR(likelihood, het_noise, uncertain_inputs, LB, _LBi_Lmi_psi1Vf, DBi_plus_BiPBi, Lm, A, psi0, psi1, beta, data_fit, num_data, output_dim, trYYT, Y, VVT_factr=None):
# the partial derivative vector for the likelihood
if likelihood.size == 0:
# save computation here.
dL_dR = None
elif het_noise:
if uncertain_inputs:
raise NotImplementedError("heteroscedatic derivates with uncertain inputs not implemented")
else:
#from ...util.linalg import chol_inv
#LBi = chol_inv(LB)
LBi, _ = dtrtrs(LB,np.eye(LB.shape[0]))
Lmi_psi1, nil = dtrtrs(Lm, psi1.T, lower=1, trans=0)
_LBi_Lmi_psi1, _ = dtrtrs(LB, Lmi_psi1, lower=1, trans=0)
dL_dR = -0.5 * beta + 0.5 * VVT_factr**2
dL_dR += 0.5 * output_dim * (psi0 - np.sum(Lmi_psi1**2,0))[:,None] * beta**2
dL_dR += 0.5*np.sum(mdot(LBi.T,LBi,Lmi_psi1)*Lmi_psi1,0)[:,None]*beta**2
dL_dR += -np.dot(_LBi_Lmi_psi1Vf.T,_LBi_Lmi_psi1).T * Y * beta**2
dL_dR += 0.5*np.dot(_LBi_Lmi_psi1Vf.T,_LBi_Lmi_psi1).T**2 * beta**2
else:
# likelihood is not heteroscedatic
dL_dR = -0.5 * num_data * output_dim * beta + 0.5 * trYYT * beta ** 2
dL_dR += 0.5 * output_dim * (psi0.sum() * beta ** 2 - np.trace(A) * beta)
dL_dR += beta * (0.5 * np.sum(A * DBi_plus_BiPBi) - data_fit)
return dL_dR
def _compute_log_marginal_likelihood(likelihood, num_data, output_dim, beta, het_noise, psi0, A, LB, trYYT, data_fit, Y):
#compute log marginal likelihood
if het_noise:
lik_1 = -0.5 * num_data * output_dim * np.log(2. * np.pi) + 0.5 * output_dim * np.sum(np.log(beta)) - 0.5 * np.sum(beta.ravel() * np.square(Y).sum(axis=-1))
lik_2 = -0.5 * output_dim * (np.sum(beta.flatten() * psi0) - np.trace(A))
else:
lik_1 = -0.5 * num_data * output_dim * (np.log(2. * np.pi) - np.log(beta)) - 0.5 * beta * trYYT
lik_2 = -0.5 * output_dim * (np.sum(beta * psi0) - np.trace(A))
lik_3 = -output_dim * (np.sum(np.log(np.diag(LB))))
lik_4 = 0.5 * data_fit
log_marginal = lik_1 + lik_2 + lik_3 + lik_4
return log_marginal