A Fast Noniterative Algorithm for Compressive Sensing Using Binary Measurement Matrices

TL;DR

This paper introduces a rapid, noniterative compressive sensing algorithm utilizing binary measurement matrices, enabling exact recovery of ultra sparse vectors in a single pass, significantly faster than traditional iterative methods.

Contribution

The paper presents a novel noniterative algorithm for compressive sensing with binary matrices, achieving exact recovery efficiently and handling nearly sparse and noisy signals.

Findings

Achieves exact recovery of ultra sparse vectors in a single pass

Significantly faster than iterative methods like $ ext{l}_1$-norm minimization

Requires fewer measurements than guaranteed recovery methods

Abstract

In this paper we present a new algorithm for compressive sensing that makes use of binary measurement matrices and achieves exact recovery of ultra sparse vectors, in a single pass and without any iterations. Due to its noniterative nature, our algorithm is hundreds of times faster than $\ell_1$-norm minimization, and methods based on expander graphs, both of which require multiple iterations. Our algorithm can accommodate nearly sparse vectors, in which case it recovers index set of the largest components, and can also accommodate burst noise measurements. Compared to compressive sensing methods that are guaranteed to achieve exact recovery of all sparse vectors, our method requires fewer measurements However, methods that achieve statistical recovery, that is, recovery of almost all but not all sparse vectors, can require fewer measurements than our method.