Approximating mixed H\"older functions using random samples

TL;DR

This paper presents a method for approximating mixed H"older functions on the unit square from random samples, achieving a specific error bound with high probability, useful for high-dimensional function approximation.

Contribution

The paper introduces a new approximation technique for mixed H"older functions based on random sampling, with proven error bounds and probabilistic guarantees.

Findings

Achieves an $L^2$ approximation error of $O(n^{-\alpha} \, ext{log}^{3/2} n)$

Requires at least $c_1 n \, ext{log}^2 n$ samples for the approximation

Guarantees high-probability bounds on approximation accuracy

Abstract

Suppose $f : [0,1]^2 \rightarrow \mathbb{R}$ is a $(c,\alpha)$-mixed H\"older function that we sample at $l$ points $X_1,\ldots,X_l$ chosen uniformly at random from the unit square. Let the location of these points and the function values $f(X_1),\ldots,f(X_l)$ be given. If $l \ge c_1 n \log^2 n$, then we can compute an approximation $\tilde{f}$ such that $$ \|f - \tilde{f} \|_{L^2} = \mathcal{O}(n^{-\alpha} \log^{3/2} n), $$ with probability at least $1 - n^{2 -c_1}$, where the implicit constant only depends on the constants $c > 0$ and $c_1 > 0$.