Random points are good for universal discretization

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.