On the construction of sparse matrices from expander graphs

TL;DR

This paper improves the theoretical understanding of constructing sparse matrices from expander graphs, reducing sample complexity and demonstrating their effectiveness in sparse recovery tasks.

Contribution

It provides new bounds on the number of nonzeros per column and quantitative sampling theorems, enhancing the construction and analysis of expander graph-based matrices.

Findings

Reduced sample complexity for matrix construction.

Outperforms existing methods in sampling theorems.

Improves sparse recovery performance with new matrices.

Abstract

We revisit the asymptotic analysis of probabilistic construction of adjacency matrices of expander graphs proposed in [4]. With better bounds we derived a new reduced sample complexity for the number of nonzeros per column of these matrices, precisely $d = \mathcal{O}\left(\log_s(N/s) \right)$; as opposed to the standard $d = \mathcal{O}\left(\log(N/s) \right)$. This gives insights into why using small $d$ performed well in numerical experiments involving such matrices. Furthermore, we derive quantitative sampling theorems for our constructions which show our construction outperforming the existing state-of-the-art. We also used our results to compare performance of sparse recovery algorithms where these matrices are used for linear sketching.