Coding-Theoretic Methods for Sparse Recovery

TL;DR

This paper explores the deep connections between coding theory and sparse recovery, showing how various combinatorial objects can be transformed into each other to improve understanding and bounds in compressed sensing and related fields.

Contribution

It introduces new reductions and concepts, such as minimum L-wise distance, linking coding properties to RIP-2 matrices and list-decoding, unifying and extending existing theories.

Findings

Unified framework for coding and sparse recovery objects

Introduction of minimum L-wise distance concept

New bounds and relationships between coding and compressed sensing

Abstract

We review connections between coding-theoretic objects and sparse learning problems. In particular, we show how seemingly different combinatorial objects such as error-correcting codes, combinatorial designs, spherical codes, compressed sensing matrices and group testing designs can be obtained from one another. The reductions enable one to translate upper and lower bounds on the parameters attainable by one object to another. We survey some of the well-known reductions in a unified presentation, and bring some existing gaps to attention. New reductions are also introduced; in particular, we bring up the notion of minimum "L-wise distance" of codes and show that this notion closely captures the combinatorial structure of RIP-2 matrices. Moreover, we show how this weaker variation of the minimum distance is related to combinatorial list-decoding properties of codes.