Infinite-Variate $L^2$-Approximation with Nested Subspace Sampling

TL;DR

This paper investigates $L^2$-approximation in infinite-dimensional weighted reproducing kernel Hilbert spaces, providing optimal algorithms and error bounds for both ANOVA and non-ANOVA spaces, highlighting differences in convergence rates.

Contribution

It introduces optimal algorithms and error bounds for $L^2$-approximation in infinite-dimensional spaces, distinguishing between ANOVA and non-ANOVA spaces with new convergence rate insights.

Findings

Optimal algorithms for ANOVA spaces with linear information.

Matching upper and lower error bounds for polynomial convergence rates.

Higher convergence rates in ANOVA spaces compared to non-ANOVA spaces for regular weights.

Abstract

We consider $L^2$-approximation on weighted reproducing kernel Hilbert spaces of functions depending on infinitely many variables. We focus on unrestricted linear information, admitting evaluations of arbitrary continuous linear functionals. We distinguish between ANOVA and non-ANOVA spaces, where, by ANOVA spaces, we refer to function spaces whose norms are induced by an underlying ANOVA function decomposition. In ANOVA spaces, we provide an optimal algorithm to solve the approximation problem using linear information. We determine the upper and lower error bounds on the polynomial convergence rate of $n$-th minimal worst-case errors, which match if the weights decay regularly. For non-ANOVA spaces, we also establish upper and lower error bounds. Our analysis reveals that for weights with a regular and moderate decay behavior, the convergence rate of $n$-th minimal errors is strictly higher in ANOVA than in non-ANOVA spaces.