# Source code for pyro.contrib.gp.models.sgpr

# Copyright (c) 2017-2019 Uber Technologies, Inc.

import torch
from torch.distributions import constraints
from torch.nn import Parameter

import pyro
import pyro.distributions as dist
from pyro.contrib.gp.models.model import GPModel
from pyro.nn.module import PyroParam, pyro_method

[docs]class SparseGPRegression(GPModel):
u"""
Sparse Gaussian Process Regression model.

In :class:.GPRegression model, when the number of input data :math:X is large,
the covariance matrix :math:k(X, X) will require a lot of computational steps to
compute its inverse (for log likelihood and for prediction). By introducing an
additional inducing-input parameter :math:X_u, we can reduce computational cost
by approximate :math:k(X, X) by a low-rank Nymstr\u00F6m approximation :math:Q
(see reference ), where

.. math:: Q = k(X, X_u) k(X,X)^{-1} k(X_u, X).

Given inputs :math:X, their noisy observations :math:y, and the inducing-input
parameters :math:X_u, the model takes the form:

.. math::
u & \\sim \\mathcal{GP}(0, k(X_u, X_u)),\\\\
f & \\sim q(f \\mid X, X_u) = \\mathbb{E}_{p(u)}q(f\\mid X, X_u, u),\\\\
y & \\sim f + \\epsilon,

where :math:\\epsilon is Gaussian noise and the conditional distribution
:math:q(f\\mid X, X_u, u) is an approximation of

.. math:: p(f\\mid X, X_u, u) = \\mathcal{N}(m, k(X, X) - Q),

whose terms :math:m and :math:k(X, X) - Q is derived from the joint
multivariate normal distribution:

.. math:: [f, u] \\sim \\mathcal{GP}(0, k([X, X_u], [X, X_u])).

This class implements three approximation methods:

+ Deterministic Training Conditional (DTC):

.. math:: q(f\\mid X, X_u, u) = \\mathcal{N}(m, 0),

which in turns will imply

.. math:: f \\sim \\mathcal{N}(0, Q).

+ Fully Independent Training Conditional (FITC):

.. math:: q(f\\mid X, X_u, u) = \\mathcal{N}(m, diag(k(X, X) - Q)),

which in turns will correct the diagonal part of the approximation in DTC:

.. math:: f \\sim \\mathcal{N}(0, Q + diag(k(X, X) - Q)).

+ Variational Free Energy (VFE), which is similar to DTC but has an additional
trace_term in the model's log likelihood. This additional term makes "VFE"
equivalent to the variational approach in :class:.SparseVariationalGP
(see reference ).

.. note:: This model has :math:\\mathcal{O}(NM^2) complexity for training,
:math:\\mathcal{O}(NM^2) complexity for testing. Here, :math:N is the number
of train inputs, :math:M is the number of inducing inputs.

References:

 A Unifying View of Sparse Approximate Gaussian Process Regression,
Joaquin Qui\u00F1onero-Candela, Carl E. Rasmussen

 Variational learning of inducing variables in sparse Gaussian processes,
Michalis Titsias

:param torch.Tensor X: A input data for training. Its first dimension is the number
of data points.
:param torch.Tensor y: An output data for training. Its last dimension is the
number of data points.
:param ~pyro.contrib.gp.kernels.kernel.Kernel kernel: A Pyro kernel object, which
is the covariance function :math:k.
:param torch.Tensor Xu: Initial values for inducing points, which are parameters
of our model.
:param torch.Tensor noise: Variance of Gaussian noise of this model.
:param callable mean_function: An optional mean function :math:m of this Gaussian
process. By default, we use zero mean.
:param str approx: One of approximation methods: "DTC", "FITC", and "VFE"
(default).
:param float jitter: A small positive term which is added into the diagonal part of
a covariance matrix to help stablize its Cholesky decomposition.
:param str name: Name of this model.
"""
def __init__(self, X, y, kernel, Xu, noise=None, mean_function=None, approx=None, jitter=1e-6):
super().__init__(X, y, kernel, mean_function, jitter)

self.Xu = Parameter(Xu)

noise = self.X.new_tensor(1.) if noise is None else noise
self.noise = PyroParam(noise, constraints.positive)

if approx is None:
self.approx = "VFE"
elif approx in ["DTC", "FITC", "VFE"]:
self.approx = approx
else:
raise ValueError("The sparse approximation method should be one of "
"'DTC', 'FITC', 'VFE'.")

[docs]    @pyro_method
def model(self):
self.set_mode("model")

# W = (inv(Luu) @ Kuf).T
# Qff = Kfu @ inv(Kuu) @ Kuf = W @ W.T
# Fomulas for each approximation method are
# DTC:  y_cov = Qff + noise,                   trace_term = 0
# FITC: y_cov = Qff + diag(Kff - Qff) + noise, trace_term = 0
# VFE:  y_cov = Qff + noise,                   trace_term = tr(Kff-Qff) / noise
# y_cov = W @ W.T + D
# trace_term is added into log_prob

N = self.X.size(0)
M = self.Xu.size(0)
Kuu = self.kernel(self.Xu).contiguous()
Kuu.view(-1)[::M + 1] += self.jitter  # add jitter to the diagonal
Luu = Kuu.cholesky()
Kuf = self.kernel(self.Xu, self.X)
W = Kuf.triangular_solve(Luu, upper=False).t()

D = self.noise.expand(N)
if self.approx == "FITC" or self.approx == "VFE":
Kffdiag = self.kernel(self.X, diag=True)
Qffdiag = W.pow(2).sum(dim=-1)
if self.approx == "FITC":
D = D + Kffdiag - Qffdiag
else:  # approx = "VFE"
trace_term = (Kffdiag - Qffdiag).sum() / self.noise
trace_term = trace_term.clamp(min=0)

zero_loc = self.X.new_zeros(N)
f_loc = zero_loc + self.mean_function(self.X)
if self.y is None:
f_var = D + W.pow(2).sum(dim=-1)
return f_loc, f_var
else:
if self.approx == "VFE":
pyro.factor(self._pyro_get_fullname("trace_term"), -trace_term / 2.)

return pyro.sample(self._pyro_get_fullname("y"),
dist.LowRankMultivariateNormal(f_loc, W, D)
.expand_by(self.y.shape[:-1])
.to_event(self.y.dim() - 1),
obs=self.y)

[docs]    @pyro_method
def guide(self):
self.set_mode("guide")

[docs]    def forward(self, Xnew, full_cov=False, noiseless=True):
r"""
Computes the mean and covariance matrix (or variance) of Gaussian Process
posterior on a test input data :math:X_{new}:

.. math:: p(f^* \mid X_{new}, X, y, k, X_u, \epsilon) = \mathcal{N}(loc, cov).

.. note:: The noise parameter noise (:math:\epsilon), the inducing-point
parameter Xu, together with kernel's parameters have been learned from
a training procedure (MCMC or SVI).

:param torch.Tensor Xnew: A input data for testing. Note that
Xnew.shape[1:] must be the same as self.X.shape[1:].
:param bool full_cov: A flag to decide if we want to predict full covariance
matrix or just variance.
:param bool noiseless: A flag to decide if we want to include noise in the
prediction output or not.
:returns: loc and covariance matrix (or variance) of :math:p(f^*(X_{new}))
:rtype: tuple(torch.Tensor, torch.Tensor)
"""
self._check_Xnew_shape(Xnew)
self.set_mode("guide")

# W = inv(Luu) @ Kuf
# Ws = inv(Luu) @ Kus
# D as in self.model()
# K = I + W @ inv(D) @ W.T = L @ L.T
# S = inv[Kuu + Kuf @ inv(D) @ Kfu]
#   = inv(Luu).T @ inv[I + inv(Luu)@ Kuf @ inv(D)@ Kfu @ inv(Luu).T] @ inv(Luu)
#   = inv(Luu).T @ inv[I + W @ inv(D) @ W.T] @ inv(Luu)
#   = inv(Luu).T @ inv(K) @ inv(Luu)
#   = inv(Luu).T @ inv(L).T @ inv(L) @ inv(Luu)
# loc = Ksu @ S @ Kuf @ inv(D) @ y = Ws.T @ inv(L).T @ inv(L) @ W @ inv(D) @ y
# cov = Kss - Ksu @ inv(Kuu) @ Kus + Ksu @ S @ Kus
#     = kss - Ksu @ inv(Kuu) @ Kus + Ws.T @ inv(L).T @ inv(L) @ Ws

N = self.X.size(0)
M = self.Xu.size(0)

# TODO: cache these calculations to get faster inference

Kuu = self.kernel(self.Xu).contiguous()
Kuu.view(-1)[::M + 1] += self.jitter  # add jitter to the diagonal
Luu = Kuu.cholesky()

Kuf = self.kernel(self.Xu, self.X)

W = Kuf.triangular_solve(Luu, upper=False)
D = self.noise.expand(N)
if self.approx == "FITC":
Kffdiag = self.kernel(self.X, diag=True)
Qffdiag = W.pow(2).sum(dim=0)
D = D + Kffdiag - Qffdiag

W_Dinv = W / D
K = W_Dinv.matmul(W.t()).contiguous()
K.view(-1)[::M + 1] += 1  # add identity matrix to K
L = K.cholesky()

# get y_residual and convert it into 2D tensor for packing
y_residual = self.y - self.mean_function(self.X)
y_2D = y_residual.reshape(-1, N).t()
W_Dinv_y = W_Dinv.matmul(y_2D)

# End caching ----------

Kus = self.kernel(self.Xu, Xnew)
Ws = Kus.triangular_solve(Luu, upper=False)
pack = torch.cat((W_Dinv_y, Ws), dim=1)
Linv_pack = pack.triangular_solve(L, upper=False)
# unpack
Linv_W_Dinv_y = Linv_pack[:, :W_Dinv_y.shape]
Linv_Ws = Linv_pack[:, W_Dinv_y.shape:]

C = Xnew.size(0)
loc_shape = self.y.shape[:-1] + (C,)
loc = Linv_W_Dinv_y.t().matmul(Linv_Ws).reshape(loc_shape)

if full_cov:
Kss = self.kernel(Xnew).contiguous()
if not noiseless:
Kss.view(-1)[::C + 1] += self.noise  # add noise to the diagonal
Qss = Ws.t().matmul(Ws)
cov = Kss - Qss + Linv_Ws.t().matmul(Linv_Ws)
cov_shape = self.y.shape[:-1] + (C, C)
cov = cov.expand(cov_shape)
else:
Kssdiag = self.kernel(Xnew, diag=True)
if not noiseless:
Kssdiag = Kssdiag + self.noise
Qssdiag = Ws.pow(2).sum(dim=0)
cov = Kssdiag - Qssdiag + Linv_Ws.pow(2).sum(dim=0)
cov_shape = self.y.shape[:-1] + (C,)
cov = cov.expand(cov_shape)

return loc + self.mean_function(Xnew), cov