A Matrix-free Approach for Solving the Parametric Gaussian Process Maximum Likelihood Problem

Authors:
Mihai Anitescu;Jie Chen;Lei Wang
Affiliations:
anitescu@mcs.anl.gov and jiechen@mcs.anl.gov and lwang@mcs.anl.gov;-;-
Venue:
SIAM Journal on Scientific Computing
Year:
2012

Citing 10
Cited 2

Optimal and superoptimal circulant preconditioners

SIAM Journal on Matrix Analysis and Applications
Variable Block CG Algorithms for Solving Large Sparse Symmetric Positive Definite Linear Systems on Parallel Computers, I: General Iterative Scheme

SIAM Journal on Matrix Analysis and Applications
A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow

Journal of Computational Physics
Combining Field Data and Computer Simulations for Calibration and Prediction

SIAM Journal on Scientific Computing
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)

Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
Fast Radial Basis Function Interpolation via Preconditioned Krylov Iteration

SIAM Journal on Scientific Computing
An Introduction to Iterative Toeplitz Solvers (Fundamentals of Algorithms)

An Introduction to Iterative Toeplitz Solvers (Fundamentals of Algorithms)
A Cartesian treecode for screened coulomb interactions

Journal of Computational Physics
Computing $f(A)b$ via Least Squares Polynomial Approximations

SIAM Journal on Scientific Computing
Fast Evaluation of Multiquadric RBF Sums by a Cartesian Treecode

SIAM Journal on Scientific Computing

Accelerating data uncertainty quantification by solving linear systems with multiple right-hand sides

Numerical Algorithms
Parameter estimation in high dimensional Gaussian distributions

Statistics and Computing

Quantified Score

Hi-index	0.00

Visualization

Abstract

Gaussian processes are the cornerstone of statistical analysis in many application areas. 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 matrix-free approach for computing 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.