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

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.