Fast Direct Methods for Gaussian Processes

A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the
$n$
-dimensional setting, however, it requires the inversion of an
$n \times n$
covariance matrix,
$C$
, as well as the evaluation of its determinant,
$\det (C)$
. In many cases, such as regression using Gaussian processes, the covariance matrix is of the form
$C = \sigma ^2 I + K$
, where
$K$
is computed using a specified covariance kernel which depends on the data and additional parameters (hyperparameters). The matrix
$C$
is typically dense, causing standard direct methods for inversion and determinant evaluation to require
$\mathcal {O}(n^3)$
work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix
$C$
can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an
$\mathcal {O} (n\,\log^2\, n)$
algorithm for inversion. More importantly, we show that this factorization enables the evaluation of the determinant
$\det (C)$
, permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining
$K$
. Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels.