A determinant-free method to simulate the parameters of large Gaussian fields

TL;DR

This paper introduces a determinant-free Bayesian inference method for high-dimensional Gaussian models, enabling efficient simulation without computing difficult determinants, suitable for large-scale applications.

Contribution

It presents a novel auxiliary-variable approach that avoids determinant calculations, improving scalability for high-dimensional Gaussian field simulations.

Findings

Efficient MCMC scheme requiring only inverse-matrix-square-roots and linear solves.

Applicable to large-scale Gaussian processes and Markov random fields.

Demonstrated effectiveness on synthetic and real-world data.

Abstract

We propose a determinant-free approach for simulation-based Bayesian inference in high-dimensional Gaussian models. We introduce auxiliary variables with covariance equal to the inverse covariance of the model. The joint probability of the auxiliary model can be computed without evaluating determinants, which are often hard to compute in high dimensions. We develop a Markov chain Monte Carlo sampling scheme for the auxiliary model that requires no more than the application of inverse-matrix-square-roots and the solution of linear systems. These operations can be performed at large scales with rational approximations. We provide an empirical study on both synthetic and real-world data for sparse Gaussian processes and for large-scale Gaussian Markov random fields.