On the Compressed Measurements over Finite Fields: Sparse or Dense Sampling

TL;DR

This paper explores the requirements for compressed sampling over finite fields, revealing that for signals that are not extremely sparse, sensing matrices can be sparse rather than dense, depending on the finite field size.

Contribution

It provides new insights into the measurement requirements and matrix sparsity conditions for successful L0 recovery over finite fields.

Findings

Sparse sensing matrices suffice for non-ultra sparse signals

Measurement bounds depend on finite field size

Dense matrices are unnecessary for certain sparsity levels

Abstract

We consider compressed sampling over finite fields and investigate the number of compressed measurements needed for successful L0 recovery. Our results are obtained while the sparseness of the sensing matrices as well as the size of the finite fields are varied. One of interesting conclusions includes that unless the signal is "ultra" sparse, the sensing matrices do not have to be dense.