Sparse recovery properties of discrete random matrices

TL;DR

This paper analyzes the linear independence properties of random ±1 matrices in the context of compressed sensing, identifying threshold behaviors for the number of columns relative to rows where independence is likely or unlikely.

Contribution

It establishes precise probabilistic thresholds for when all subsets of a certain size of columns are linearly independent in random ±1 matrices.

Findings

For d ≤ n^{1+1/2(1-δ)-o(1)}, all s-column subsets are linearly independent with high probability.

For d ≥ n^{1+1/2(1-δ)+o(1)}, some s-column subsets are linearly dependent with high probability.

Threshold behavior depends on the relation between d and n, with sharp phase transitions.

Abstract

Motivated by problems from compressed sensing, we determine the threshold behavior of a random $n\times d$ $\pm 1$ matrix $M_{n,d}$ with respect to the property "every $s$ columns are linearly independent". In particular, we show that for every $0<\delta <1$ and $s=(1-\delta)n$, if $d\leq n^{1+1/2(1-\delta)-o(1)}$ then with high probability every $s$ columns of $M_{n,d}$ are linearly independent, and if $d\geq n^{1+1/2(1-\delta)+o(1)}$ then with high probability there are some $s$ linearly dependent columns.