The covariance matrix of Green's functions and its application to machine learning

TL;DR

This paper introduces a Green's function-based regression algorithm utilizing the covariance matrix of Green's functions, providing a Bayesian predictive distribution with confidence intervals for machine learning tasks.

Contribution

It presents a novel regression method leveraging Green's functions and their covariance matrix, integrating Bayesian inference for improved predictions.

Findings

Provides a Green's function-based regression algorithm.

Derives a covariance matrix as a probability density function.

Offers predictive mean and confidence intervals in Bayesian framework.

Abstract

In this paper, a regression algorithm based on Green's function theory is proposed and implemented. We first survey Green's function for the Dirichlet boundary value problem of 2nd order linear ordinary differential equation, which is a reproducing kernel of a suitable Hilbert space. We next consider a covariance matrix composed of the normalized Green's function, which is regarded as aprobability density function. By supporting Bayesian approach, the covariance matrix gives predictive distribution, which has the predictive mean $\mu$ and the confidence interval [$\mu$-2s, $\mu$+2s], where s stands for a standard deviation.