Bounds for the sampling discretization error and their applications to the universal sampling discretization

TL;DR

The paper addresses the problem of discretizing the $L_p$ norm over function classes by establishing two general upper bounds for the discretization error $er_m(W,L_p)$ that relate to entropy numbers in the uniform norm via generic chaining and Lipschitz transforms. It then applies these bounds to classes with small mixed smoothness, deriving explicit rates such as $er_m(\mathbf W^r_q,L_p)\lesssim m^{-r}(\log m)^{(d-1)(1-r)+r}$ and $er_m(\mathbf H^r_q,L_p)\lesssim m^{-r}(\log m)^{d-1+r}$, with refined forms at $r=1/2$. A second bound using a GMPT-type framework yields tail bounds for the distribution of the discretization error and connects entropy bounds of transformed classes $G=\{|f|^{p/2}: f\in W\}$ to $er_m$. The paper also develops universal sampling discretization results for families of finite-dimensional subspaces, providing explicit sample-size bounds $m$ (scaling like $Kv\log N$ with polylog factors) to achieve $L_p$-usd for all subspaces up to dimension $v$, and extends these results to $p\in(0,2]$ with improved dependence on the approximation parameter. Overall, the work gives rigorous, quantitative guarantees for both classical discretization and universal discretization across a broad range of smoothness classes and $p$.

Abstract

In the first part of the paper we study absolute error of sampling discretization of the integral $L_p$-norm for function classes of continuous functions. We use basic approaches from chaining technique to provide general upper bounds for the error of sampling discretization of the $L_p$-norm on a given function class in terms of entropy numbers in the uniform norm of this class. As an example we apply these general results to obtain new error bounds for sampling discretization of the $L_p$-norms on classes of multivariate functions with mixed smoothness. In the second part of the paper we apply our general bounds to study the problem of universal sampling discretization.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.1
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.1: see Theorem 2.2.18 in Tal