Lengthscale-informed sparse grids for kernel methods in high dimensions

TL;DR

This paper introduces a novel sparse grid method that leverages lengthscale parameters in Matérn kernels to improve high-dimensional kernel interpolation by exploiting functional anisotropy, reducing the curse of dimensionality.

Contribution

It generalizes sparse grid techniques by incorporating kernel lengthscale information to better handle anisotropic functions in high dimensions.

Findings

Effective emulation in high dimensions for anisotropic functions

Error bounds demonstrating improved approximation

Numerical experiments confirming practical benefits

Abstract

Kernel interpolation, especially in the context of Gaussian process emulation, is a widely used technique in surrogate modelling, where the goal is to cheaply approximate an input-output map using a limited number of function evaluations. However, in high-dimensional settings, such methods typically suffer from the curse of dimensionality; the number of required evaluations to achieve a fixed approximation error grows exponentially with the input dimension. To overcome this, a common technique used in high-dimensional approximation methods, such as quasi-Monte Carlo and sparse grids, is to exploit functional anisotropy: the idea that some input dimensions are more 'sensitive' than others. In doing so, such methods can significantly reduce the dimension dependence in the error. In this work, we propose a generalisation of sparse grid methods that incorporates a form of anisotropy encoded by the lengthscale parameter in Mat\'ern kernels. We derive error bounds and perform numerical experiments that show that our approach enables effective emulation over arbitrarily high dimensions for functions exhibiting sufficient anisotropy.