Random points are good for universal discretization

TL;DR

This paper demonstrates that random points can effectively achieve universal discretization of integral norms with high probability, and introduces a greedy algorithm for sparse recovery based on these points.

Contribution

It shows that i.i.d. random points can provide good universal discretization with high probability and proposes a simple greedy algorithm for sparse recovery.

Findings

Random i.i.d. points achieve universal discretization with high probability.

A greedy algorithm based on these points yields excellent sparse recovery results.

The approach enhances sampling recovery methods in various applications.

Abstract

There has been significant progress in the study of sampling discretization of integral norms for both a designated finite-dimensional function space and a finite collection of such function spaces (universal discretization). Sampling discretization results turn out to be very useful in various applications, particularly in sampling recovery. Recent sampling discretization results typically provide existence of good sampling points for discretization. In this paper, we show that independent and identically distributed random points provide good universal discretization with high probability. Furthermore, we demonstrate that a simple greedy algorithm based on those points that are good for universal discretization provides excellent sparse recovery results in the square norm.