Coordinate restrictions of linear operators in $l_2^n$

TL;DR

This paper explores how restricting linear operators to coordinate subspaces in high-dimensional spaces can improve their properties, with applications in harmonic analysis and data transmission networks.

Contribution

It introduces new probabilistic methods for operator restriction, extending existing theorems and providing n-independent results based on the Hilbert-Schmidt norm.

Findings

Probabilistic extension of Kashin and Tzafriri's suppression theorem

New approach to Rudelson's result on random vectors in isotropic position

Generalization of Bourgain-Tzafriri's invertibility principle

Abstract

This paper addresses the problem of improving properties of a linear operator u in $l_2^n$ by restricting it onto coordinate subspaces. We discuss how to reduce the norm of u by a random coordinate restriction, how to approximate u by a random operator with small "coordinate" rank, how to find coordinate subspaces where u is an isomorphism. The first problem in this list provides a probabilistic extension of a suppression theorem of Kashin and Tzafriri, the second one is a new look at a result of Rudelson on the random vectors in the isotropic position, the last one is the recent generalization of the Bourgain-Tzafriri's invertibility principle. The main point is that all the results are independent of n, the situation is instead controlled by the Hilbert-Schmidt norm of u. As an application, we provide an almost optimal solution to the problem of harmonic density in harmonic analysis, and a solution to the reconstruction problem for communication networks which deliver data with random losses.