An algebraic perspective on integer sparse recovery

TL;DR

This paper investigates integer sparse recovery in compressed sensing, exploring how additional lattice-valued structure affects sampling and reconstruction, and providing theoretical insights and constructions using combinatorial, probabilistic, and number-theoretic methods.

Contribution

It introduces a new algebraic perspective on integer sparse recovery, analyzing when extra lattice structure can improve sampling and reconstruction guarantees.

Findings

Existence of sensing matrices with integer structure

Limitations of the algebraic framework demonstrated by Minkowski theorems

Concrete examples of sampling designs for integer sparse signals

Abstract

Compressed sensing is a relatively new mathematical paradigm that shows a small number of linear measurements are enough to efficiently reconstruct a large dimensional signal under the assumption the signal is sparse. Applications for this technology are ubiquitous, ranging from wireless communications to medical imaging, and there is now a solid foundation of mathematical theory and algorithms to robustly and efficiently reconstruct such signals. However, in many of these applications, the signals of interest do not only have a sparse representation, but have other structure such as lattice-valued coefficients. While there has been a small amount of work in this setting, it is still not very well understood how such extra information can be utilized during sampling and reconstruction. Here, we explore the problem of integer sparse reconstruction, lending insight into when this knowledge can be useful, and what types of sampling designs lead to robust reconstruction guarantees. We use a combination of combinatorial, probabilistic and number-theoretic methods to discuss existence and some constructions of such sensing matrices with concrete examples. We also prove sparse versions of Minkowski's Convex Body and Linear Forms theorems that exhibit some limitations of this framework.