Vanishingly Sparse Matrices and Expander Graphs, With Application to Compressed Sensing

TL;DR

This paper analyzes the properties of sparse random matrices related to expander graphs, providing probabilistic bounds and a novel dyadic splitting technique, with applications to compressed sensing.

Contribution

It introduces a detailed probabilistic analysis of sparse matrices and expander graphs, including new bounds and a dyadic splitting method for better theoretical guarantees.

Findings

Derived bounds on neighbor set cardinality and expansion probability

Introduced a novel dyadic splitting technique for analysis

Provided quantitative theorems for lossless expanders and compressed sensing

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 with replacement. These matrices have a one-to-one correspondence with the adjacency matrices of fixed left degree expander graphs. We present formulae for the expected cardinality of the set of neighbors for these graphs, and present tail bounds on the probability that this cardinality will be less than the expected value. Deducible from these bounds are similar bounds for the expansion of the graph which is of interest in many applications. These bounds are derived through a more detailed analysis of collisions in unions of sets. Key to this analysis is a novel {\em dyadic splitting} technique. The analysis led to the derivation of better order 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.