Source code for GPy.inference.latent_function_inference.dtc

# Copyright (c) 2012-2014, James Hensman

from .posterior import Posterior
from ...util.linalg import jitchol, tdot, dtrtrs, dpotri, pdinv
import numpy as np
from . import LatentFunctionInference
log_2_pi = np.log(2*np.pi)

[docs]class DTC(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.

NB. It's not recommended to use this function! It's here for historical purposes.

"""
def __init__(self):
self.const_jitter = 1e-6

[docs]    def inference(self, kern, X, Z, likelihood, Y, mean_function=None, Y_metadata=None):
assert mean_function is None, "inference with a mean function not implemented"
assert X_variance is None, "cannot use X_variance with DTC. Try varDTC."

num_inducing, _ = Z.shape
num_data, output_dim = Y.shape

#make sure the noise is not hetero
if precision.size > 1:
raise NotImplementedError("no hetero noise with this implementation of DTC")

Kmm = kern.K(Z)
Knn = kern.Kdiag(X)
Knm = kern.K(X, Z)
U = Knm
Uy = np.dot(U.T,Y)

#factor Kmm
Kmmi, L, Li, _ = pdinv(Kmm)

# Compute A
LiUTbeta = np.dot(Li, U.T)*np.sqrt(precision)
A = tdot(LiUTbeta) + np.eye(num_inducing)

# factor A
LA = jitchol(A)

# back substutue to get b, P, v
tmp, _ = dtrtrs(L, Uy, lower=1)
b, _ = dtrtrs(LA, tmp*precision, lower=1)
tmp, _ = dtrtrs(LA, b, lower=1, trans=1)
v, _ = dtrtrs(L, tmp, lower=1, trans=1)
tmp, _ = dtrtrs(LA, Li, lower=1, trans=0)
P = tdot(tmp.T)

#compute log marginal
log_marginal = -0.5*num_data*output_dim*np.log(2*np.pi) + \
-np.sum(np.log(np.diag(LA)))*output_dim + \
0.5*num_data*output_dim*np.log(precision) + \
-0.5*precision*np.sum(np.square(Y)) + \
0.5*np.sum(np.square(b))

# Compute dL_dKmm
vvT_P = tdot(v.reshape(-1,1)) + P
dL_dK = 0.5*(Kmmi - vvT_P)

# Compute dL_dU
vY = np.dot(v.reshape(-1,1),Y.T)
dL_dU = vY - np.dot(vvT_P, U.T)
dL_dU *= precision

#compute dL_dR
Uv = np.dot(U, v)
dL_dR = 0.5*(np.sum(U*np.dot(U,P), 1) - 1./precision + np.sum(np.square(Y), 1) - 2.*np.sum(Uv*Y, 1) + np.sum(np.square(Uv), 1))*precision**2

grad_dict = {'dL_dKmm': dL_dK, 'dL_dKdiag':np.zeros_like(Knn), 'dL_dKnm':dL_dU.T, 'dL_dthetaL':dL_dthetaL}

#construct a posterior object
post = Posterior(woodbury_inv=Kmmi-P, woodbury_vector=v, K=Kmm, mean=None, cov=None, K_chol=L)

[docs]class vDTC(object):
def __init__(self):
self.const_jitter = 1e-6

[docs]    def inference(self, kern, X, Z, likelihood, Y, mean_function=None, Y_metadata=None):
assert mean_function is None, "inference with a mean function not implemented"
assert X_variance is None, "cannot use X_variance with DTC. Try varDTC."

num_inducing, _ = Z.shape
num_data, output_dim = Y.shape

#make sure the noise is not hetero
if precision.size > 1:
raise NotImplementedError("no hetero noise with this implementation of DTC")

Kmm = kern.K(Z)
Knn = kern.Kdiag(X)
Knm = kern.K(X, Z)
U = Knm
Uy = np.dot(U.T,Y)

#factor Kmm
Kmmi, L, Li, _ = pdinv(Kmm)

# Compute A
LiUTbeta = np.dot(Li, U.T)*np.sqrt(precision)
A_ = tdot(LiUTbeta)
trace_term = -0.5*(np.sum(Knn)*precision - np.trace(A_))
A = A_ + np.eye(num_inducing)

# factor A
LA = jitchol(A)

# back substutue to get b, P, v
tmp, _ = dtrtrs(L, Uy, lower=1)
b, _ = dtrtrs(LA, tmp*precision, lower=1)
tmp, _ = dtrtrs(LA, b, lower=1, trans=1)
v, _ = dtrtrs(L, tmp, lower=1, trans=1)
tmp, _ = dtrtrs(LA, Li, lower=1, trans=0)
P = tdot(tmp.T)
stop

#compute log marginal
log_marginal = -0.5*num_data*output_dim*np.log(2*np.pi) + \
-np.sum(np.log(np.diag(LA)))*output_dim + \
0.5*num_data*output_dim*np.log(precision) + \
-0.5*precision*np.sum(np.square(Y)) + \
0.5*np.sum(np.square(b)) + \
trace_term

# Compute dL_dKmm
vvT_P = tdot(v.reshape(-1,1)) + P
LAL = Li.T.dot(A).dot(Li)
dL_dK = Kmmi - 0.5*(vvT_P + LAL)

# Compute dL_dU
vY = np.dot(v.reshape(-1,1),Y.T)
#dL_dU = vY - np.dot(vvT_P, U.T)
dL_dU = vY - np.dot(vvT_P - Kmmi, U.T)
dL_dU *= precision

#compute dL_dR
Uv = np.dot(U, v)
dL_dR = 0.5*(np.sum(U*np.dot(U,P), 1) - 1./precision + np.sum(np.square(Y), 1) - 2.*np.sum(Uv*Y, 1) + np.sum(np.square(Uv), 1) )*precision**2
dL_dR -=precision*trace_term/num_data