Universal sampling discretization

TL;DR

This paper establishes improved universal sampling discretization bounds for integral norms of functions in high-dimensional subspaces, significantly reducing the number of required sampling points compared to previous results.

Contribution

It introduces a new bound on the number of sampling points needed for discretization, which is much smaller than prior quadratic bounds, using entropy number inequalities and greedy approximation.

Findings

Discretization bound $m \\ll v(\\\log N)^2(\\\log v)^2$

Improved over previous quadratic bounds in $v$

Uses entropy numbers and greedy approximation techniques

Abstract

Let $X_N$ be an $N$-dimensional subspace of $L_2$ functions on a probability space $(\Omega, \mu)$ spanned by a uniformly bounded Riesz basis $\Phi_N$. Given an integer $1\leq v\leq N$ and an exponent $1\leq q\leq 2$, we obtain universal discretization for integral norms $L_q(\Omega,\mu)$ of functions from the collection of all subspaces of $X_N$ spanned by $v$ elements of $\Phi_N$ with the number $m$ of required points satisfying $m\ll v(\log N)^2(\log v)^2$. This last bound on $m$ is much better than previously known bounds which are quadratic in $v$. Our proof uses a conditional theorem on universal sampling discretization, and an inequality of entropy numbers in terms of greedy approximation with respect to dictionaries.