Statistical Mechanical Analysis of Gaussian Processes

TL;DR

This paper applies statistical mechanics to analyze Gaussian processes by simplifying inputs to one dimension and using periodic boundary conditions, enabling diagonalization of the covariance matrix and matching analytical solutions with simulations.

Contribution

It introduces a novel approach to analyze Gaussian processes through statistical mechanics with simplified models and boundary conditions, facilitating analytical solutions.

Findings

Analytical solutions closely match simulation results.

Diagonalization of covariance matrix simplifies analysis.

Periodic boundary conditions are effective in modeling Gaussian processes.

Abstract

In this paper, we analyze Gaussian processes using statistical mechanics. Although the input is originally multidimensional, we simplify our model by considering the input as one-dimensional for statistical mechanical analysis. Furthermore, we employ periodic boundary conditions as an additional modeling approach. By using periodic boundary conditions, we can diagonalize the covariance matrix. The diagonalized covariance matrix is then applied to Gaussian processes. This allows for a statistical mechanical analysis of Gaussian processes using the derived diagonalized matrix. We indicate that the analytical solutions obtained in this method closely match the results from simulations.