On construction and analysis of sparse random matrices and expander graphs with applications to compressed sensing

TL;DR

This paper analyzes the probabilistic construction of sparse random matrices and their associated expander graphs, providing new tail bounds and constants that improve understanding of their properties and applications in compressed sensing.

Contribution

It introduces improved tail bounds and constants for sparse random matrices and expander graphs, enhancing theoretical guarantees in compressed sensing.

Findings

Derived better tail bounds for neighbor set cardinality

Established quantitative theorems for lossless expander graphs

Provided improved constants for compressed sensing sampling

Abstract

We revisit the probabilistic construction of sparse random matrices where each column has a fixed number of nonzeros whose row indices are drawn uniformly at random. These matrices have a one-to-one correspondence with the adjacency matrices of lossless expander graphs. We present tail bounds on the probability that the cardinality of the set of neighbors for these graphs will be less than the expected value. The bounds are derived through the analysis of collisions in unions of sets using a {\em dyadic splitting} technique. This analysis led to the derivation of better constants that allow for quantitative theorems on existence of lossless expander graphs and hence the sparse random matrices we consider and also quantitative compressed sensing sampling theorems when using sparse non mean-zero measurement matrices.