The loss of the property of locality of the kernel in high-dimensional Gaussian process regression on the example of the fitting of molecular potential energy surfaces

TL;DR

This paper investigates how the property of locality in Gaussian-like kernels diminishes in high-dimensional spaces, impacting the effectiveness of Gaussian process regression for molecular potential energy surface fitting.

Contribution

It demonstrates the practical loss of kernel locality in high dimensions and introduces a multi-zeta kernel approach, analyzing its performance across different dimensionalities.

Findings

Locality of Gaussian-like kernels diminishes in high-dimensional spaces

Multi-zeta kernel improves regression in low dimensions

Loss of locality reduces the advantage of multi-zeta kernels in high dimensions

Abstract

Kernel based methods including Gaussian process regression (GPR) and generally kernel ridge regression (KRR) have been finding increasing use in computational chemistry, including the fitting of potential energy surfaces and density functionals in high-dimensional feature spaces. Kernels of the Matern family such as Gaussian-like kernels (basis functions) are often used, which allows imparting them the meaning of covariance functions and formulating GPR as an estimator of the mean of a Gaussian distribution. The notion of locality of the kernel is critical for this interpretation. It is also critical to the formulation of multi-zeta type basis functions widely used in computational chemistry We show, on the example of fitting of molecular potential energy surfaces of increasing dimensionality, the practical disappearance of the property of locality of a Gaussian-like kernel in high dimensionality. We also formulate a multi-zeta approach to the kernel and show that it significantly improves the quality of regression in low dimensionality but loses any advantage in high dimensionality, which is attributed to the loss of the property of locality.