 
Summary: Preprint ANL/MCSP18570311
A MATRIXFREE APPROACH FOR SOLVING THE GAUSSIAN
PROCESS MAXIMUM LIKELIHOOD PROBLEM
MIHAI ANITESCU, JIE CHEN, AND LEI WANG
Abstract. Gaussian processes are the cornerstone of statistical analysis in many application ar
eas. Nevertheless, most of the applications are limited by their need to use the Cholesky factorization
in the computation of the likelihood. In this work, we present a matrixfree approach for comput
ing the solution of the maximum likelihood problem involving Gaussian processes. The approach is
based on a stochastic programming reformulation followed by sample average approximation applied
to either the maximization problem or its optimality conditions. We provide statistical estimates of
the approximate solution. The method is illustrated on several examples where the data is provided
on a regular or irregular grid. In the latter case, the action of a covariance matrix on a vector is
computed by means of fast multipole methods. For each of the examples presented, we demonstrate
that the approach scales linearly with an increase in the number of sites.
Key words. Gaussian process, maximum likelihood estimation, sample average approximation,
preconditioned conjugate gradient, Toeplitz system, circulant preconditioner, fast multipole method
AMS subject classifications. 65C60, 90C15, 65F10, 65F30
1. Introduction. Gaussian processes (GPs) have been widely used throughout
the statistical and machine learning communities for modeling natural processes [6, 7]
for regression and classification problems [26], and for interpolation in parameter
