Randomized mixed H\"older function approximation in higher-dimensions

TL;DR

This paper extends previous results to higher-dimensional mixed H"older functions, showing that sampling at a specific rate yields accurate approximations with high probability.

Contribution

It generalizes mixed H"older function approximation results to all dimensions, providing explicit sampling complexity and error bounds.

Findings

Sampling at ~ (1/ε)(log(1/ε))^d points achieves desired accuracy.

Approximation error is bounded by ε^α times a logarithmic factor.

High probability guarantees for the approximation quality.

Abstract

The purpose of this paper is to extend the result of arXiv:1810.00823 to mixed H\"older functions on $[0,1]^d$ for all $d \ge 1$. In particular, we prove that by sampling an $\alpha$-mixed H\"older function $f : [0,1]^d \rightarrow \mathbb{R}$ at $\sim \frac{1}{\varepsilon} \left(\log \frac{1}{\varepsilon} \right)^d$ independent uniformly random points from $[0,1]^d$, we can construct an approximation $\tilde{f}$ such that $$ \|f - \tilde{f}\|_{L^2} \lesssim \varepsilon^\alpha \left(\log \textstyle{\frac{1}{\varepsilon}} \right)^{d-1/2}, $$ with high probability.