On the Theoretical Guarantees for Parameter Estimation of Gaussian Random Field Models: A Sparse Precision Matrix Approach

TL;DR

This paper introduces a two-stage method for estimating Gaussian Random Field parameters that combines sparse precision matrix estimation with covariance parameter fitting, supported by theoretical error bounds.

Contribution

It proposes a novel two-stage approach using convex regularization and least squares, with theoretical guarantees for parameter estimation in GRFs.

Findings

Provides tight theoretical error bounds for both stages.

Demonstrates the effectiveness of the sparse precision matrix approach.

Addresses computational challenges in high-dimensional anisotropic GRFs.

Abstract

Iterative methods for fitting a Gaussian Random Field (GRF) model via maximum likelihood (ML) estimation requires solving a nonconvex optimization problem. The problem is aggravated for anisotropic GRFs where the number of covariance function parameters increases with the dimension. Even evaluation of the likelihood function requires $O(n^3)$ floating point operations, where $n$ denotes the number of data locations. In this paper, we propose a new two-stage procedure to estimate the parameters of second-order stationary GRFs. First, a convex likelihood problem regularized with a weighted $\ell_1$-norm, utilizing the available distance information between observation locations, is solved to fit a sparse precision (inverse covariance) matrix to the observed data. Second, the parameters of the covariance function are estimated by solving a least squares problem. Theoretical error bounds for the solutions of stage I and II problems are provided, and their tightness are investigated.