Extreme Singular Values of Random Time-Frequency Structured Matrices

TL;DR

This paper studies the extreme singular values of random Gabor frame analysis matrices, providing bounds that depend on the structure and size of the time-frequency shift set, and compares these with matrices having i.i.d. entries.

Contribution

It introduces bounds on singular values of random time-frequency structured matrices, highlighting the influence of the shift set's structure and size, a novel analysis in this context.

Findings

Derived bounds on singular values depending on the set structure

Compared structured matrices with i.i.d. matrices

Analyzed dependence on the size of the shift set

Abstract

In this paper, we investigate extreme singular values of the analysis matrix of a Gabor frame $(g, \Lambda)$ with a random window $g$. Columns of such matrices are time and frequency shifts of $g$, and $\Lambda\subset \mathbb{Z}_M\times\mathbb{Z}_M$ is the set of time-frequency shift indices. Our aim is to obtain bounds on the singular values of such random time-frequency structured matrices for various choices of the frame set $\Lambda$, and to investigate their dependence on the structure of $\Lambda$, as well as on its cardinality. We also compare the results obtained for Gabor frame analysis matrices with the respective results for matrices with independent identically distributed entries.