Local, algebraic simplifications of Gaussian random fields

TL;DR

This paper introduces a method to simplify the computation of local Taylor coefficients of Gaussian random fields with Gaussian covariance, avoiding matrix inversion, with applications in physics and machine learning.

Contribution

It provides a novel approach to compute local Taylor coefficients of Gaussian fields without matrix inversion, applicable in multiple dimensions and orders.

Findings

Enables explicit generation of complex potential landscapes.

Simplifies hyperparameter estimation in Gaussian process regression.

Applicable to high-dimensional fields and various orders.

Abstract

Many applications of Gaussian random fields and Gaussian random processes are limited by the computational complexity of evaluating the probability density function, which involves inverting the relevant covariance matrix. In this work, we show how that problem can be completely circumvented for the local Taylor coefficients of a Gaussian random field with a Gaussian (or `square exponential') covariance function. Our results hold for any dimension of the field and to any order in the Taylor expansion. We present two applications. First, we show that this method can be used to explicitly generate non-trivial potential energy landscapes with many fields. This application is particularly useful when one is concerned with the field locally around special points (e.g.~maxima or minima), as we exemplify by the problem of cosmic `manyfield' inflation in the early universe. Second, we show that this method has applications in machine learning, and greatly simplifies the regression problem of determining the hyperparameters of the covariance function given a training data set consisting of local Taylor coefficients at single point. An accompanying Mathematica notebook is available at https://doi.org/10.17863/CAM.22859 .