Asymptotic analysis in multivariate average case approximation with Gaussian kernels

TL;DR

This paper analyzes the asymptotic behavior of the average case approximation complexity for high-dimensional tensor product Gaussian kernel-based random fields, providing criteria for growth and boundedness as dimension increases.

Contribution

It establishes necessary and sufficient conditions for the asymptotic growth of approximation complexity in high dimensions, linking it to quantiles of self-decomposable distributions.

Findings

Criteria for boundedness of complexity with increasing dimension

Asymptotic formulas involving quantiles of self-decomposable distributions

Application to approximation problems with Gaussian kernels

Abstract

We consider tensor product random fields $Y_d$, $d\in\mathbb{N}$, whose covariance funtions are Gaussian kernels. The average case approximation complexity $n^{Y_d}(\varepsilon)$ is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate $Y_d$, with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. We investigate the growth of $n^{Y_d}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$. Namely, we find criteria of boundedness for $n^{Y_d}(\varepsilon)$ on $d$ and of tending $n^{Y_d}(\varepsilon)\to\infty$, $d\to\infty$, for any fixed $\varepsilon\in(0,1)$. In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptotics \begin{eqnarray*} \ln n^{Y_d}(\varepsilon)= a_d+q(\varepsilon)b_d+o(b_d),\quad d\to\infty, \end{eqnarray*} with any $\varepsilon\in(0,1)$. Here $q\colon (0,1)\to\mathbb{R}$ is a non-decreasing function, $(a_d)_{d\in\mathbb{N}}$ is a sequence and $(b_d)_{d\in\mathbb{N}}$ is a positive sequence such that $b_d\to\infty$, $d\to\infty$. We show that only special quantiles of self-decomposable distribution functions appear as functions $q$ in a given asymptotics. These general results apply to $n^{Y_d}(\varepsilon)$ under particular assumptions on the length scale parameters.