Norms of random matrices: local and global problems

TL;DR

This paper investigates how modifying a small part of a random matrix can optimize its operator norm, revealing conditions for effective improvement and employing advanced matrix norms and factorization techniques.

Contribution

It establishes the precise conditions under which zeroing out a small submatrix reduces the operator norm of a random matrix to the optimal order, using novel analytical methods.

Findings

Operator norm can be reduced to O(√n) by removing a small submatrix.

Zero mean and finite variance are necessary for norm reduction.

The size of the submatrix relates almost optimally to the achieved norm.

Abstract

Can the behavior of a random matrix be improved by modifying a small fraction of its entries? Consider a random matrix $A$ with i.i.d. entries. We show that the operator norm of $A$ can be reduced to the optimal order $O(\sqrt{n})$ by zeroing out a small submatrix of $A$ if and only if the entries have zero mean and finite variance. Moreover, we obtain an almost optimal dependence between the size of the removed submatrix and the resulting operator norm. Our approach utilizes the cut norm and Grothendieck-Pietsch factorization for matrices, and it combines the methods developed recently by C. Le and R. Vershynin and by E. Rebrova and K. Tikhomirov.