Sampling discretization and related problems

TL;DR

This survey reviews sampling discretization of norms in finite-dimensional spaces, highlighting classical and recent results, and explores its connections with spectral properties, operator norms, embeddings, and learning theory.

Contribution

It provides a comprehensive overview of sampling discretization results and techniques, emphasizing recent advances and interdisciplinary connections.

Findings

Classical and recent results on sampling discretization of integral and uniform norms.

Connections between sampling discretization and spectral properties, operator norms, and embeddings.

Discussion of key techniques used in proving sampling discretization results.

Abstract

This survey addresses sampling discretization and its connections with other areas of mathematics. The survey concentrates on sampling discretization of norms of elements of finite-dimensional subspaces. We present here known results on sampling discretization of both integral norms and the uniform norm beginning with classical results and ending with very recent achievements. We also show how sampling discretization connects to spectral properties and operator norms of submatrices, embedding of finite-dimensional subspaces, moments of marginals of high-dimensional distributions, and learning theory. Along with the corresponding results, important techniques for proving those results are discussed as well.