Asymptotic analysis in multivariate worst case approximation with Gaussian kernels

TL;DR

This paper analyzes how the complexity of approximating multivariate functions with Gaussian kernels grows with dimension, providing asymptotic results for fixed and shrinking error thresholds in a worst-case setting.

Contribution

It offers new asymptotic analysis of information complexity for high-dimensional Gaussian kernel approximation, a topic not extensively studied before.

Findings

Complexity grows exponentially with dimension for fixed error thresholds.

Asymptotic formulas are derived for both fixed and vanishing error thresholds.

Results inform the feasibility of high-dimensional Gaussian kernel approximation.

Abstract

We consider a problem of approximation of $d$-variate functions defined on $\mathbb{R}^d$ which belong to the Hilbert space with tensor product-type reproducing Gaussian kernel with constant shape parameter. Within worst case setting, we investigate the growth of the information complexity as $d\to\infty$. The asymptotics are obtained for the case of fixed error threshold and for the case when it goes to zero as $d\to\infty$.