Posterior contraction in Gaussian process regression using Wasserstein approximations

TL;DR

This paper investigates how quickly Gaussian process regression models converge to the true function in unbounded domains by approximating the posterior distribution with Wasserstein distance, providing new theoretical insights.

Contribution

It introduces a novel Wasserstein-based Gaussian approximation for the posterior in Gaussian process regression, linking approximation accuracy to contraction rates.

Findings

Wasserstein approximation effectively characterizes posterior contraction.

Results apply to Gaussian and squared-exponential kernels.

Provides theoretical bounds on convergence rates.

Abstract

We study posterior rates of contraction in Gaussian process regression with unbounded covariate domain. Our argument relies on developing a Gaussian approximation to the posterior of the leading coefficients of a Karhunen--Lo\'{e}ve expansion of the Gaussian process. The salient feature of our result is deriving such an approximation in the $L^2$ Wasserstein distance and relating the speed of the approximation to the posterior contraction rate using a coupling argument. Specific illustrations are provided for the Gaussian or squared-exponential covariance kernel.