Recovery of Sobolev functions restricted to iid sampling

TL;DR

This paper compares the effectiveness of iid sampling versus randomized algorithms for approximating and integrating Sobolev functions, showing they achieve the same convergence rates except in a specific case.

Contribution

It demonstrates that iid sampling attains the same optimal convergence rates as randomized algorithms for Sobolev function approximation, except when p=q=∞.

Findings

Same optimal convergence rates for iid sampling and randomized algorithms.

Logarithmic loss occurs only when p=q=∞.

Results applicable to learning and uncertainty quantification.

Abstract

We study $L_q$-approximation and integration for functions from the Sobolev space $W^s_p(\Omega)$ and compare optimal randomized (Monte Carlo) algorithms with algorithms that can only use iid sample points, uniformly distributed on the domain. The main result is that we obtain the same optimal rate of convergence if we restrict to iid sampling, a common assumption in learning and uncertainty quantification. The only exception is when $p=q=\infty$, where a logarithmic loss cannot be avoided.