On the Role of Sparsity in Compressed Sensing and Random Matrix Theory

TL;DR

This paper explores how sparsity concepts influence compressed sensing and random matrix theory, providing bounds on matrix invertibility and the smallest singular value.

Contribution

It introduces an argument that achieves optimal bounds on the smallest singular value of random matrices, emphasizing the role of sparsity.

Findings

Established the N^{-1/2} bound on the median smallest singular value

Highlighted the critical role of sparsity in invertibility proofs

Connected compressed sensing ideas with random matrix invertibility

Abstract

We discuss applications of some concepts of Compressed Sensing in the recent work on invertibility of random matrices due to Rudelson and the author. We sketch an argument leading to the optimal bound N^{-1/2} on the median of the smallest singular value of an N by N matrix with random independent entries. We highlight the parts of the argument where sparsity ideas played a key role.