Norms of Randomized Circulant Matrices

TL;DR

This paper studies bounds on the operator norms of random matrices, especially circulant matrices, providing new bounds and conjectures that improve understanding of their spectral properties.

Contribution

It introduces a lower bound for Rademacher matrices and conjectures a reversed bound, proving it for circulant matrices up to a double logarithmic factor.

Findings

Lower bound for Rademacher matrices established

Conjecture on reversed bounds proposed and partially validated

Bounds improved for circulant matrices under mild conditions

Abstract

We investigate two-sided bounds for operator norms of random matrices with unhomogenous independent entries. We formulate a lower bound for Rademacher matrices and conjecture that it may be reversed up to a universal constant. We show that our conjecture holds up to $\log\log n$ factor for randomized $n\times n$ circulant matrices and double logarithm may be eliminated under some mild additional assumptions on the coefficients.