A note on practical approximate projection schemes in signal space methods

TL;DR

This paper investigates the use of compressive sensing methods as approximate projection schemes in signal space methods, providing both empirical evidence and theoretical guarantees for their effectiveness with certain sparse signals.

Contribution

It offers the first theoretical analysis supporting the empirical use of CS methods as approximate projections in signal space algorithms.

Findings

CS methods can serve as effective approximate projections for certain signals.

Theoretical guarantees align with experimental results.

Applicable to signals sparse in arbitrary dictionaries.

Abstract

Compressive sensing (CS) is a new technology which allows the acquisition of signals directly in compressed form, using far fewer measurements than traditional theory dictates. Recently, many so-called signal space methods have been developed to extend this body of work to signals sparse in arbitrary dictionaries rather than orthonormal bases. In doing so, CS can be utilized in a much broader array of practical settings. Often, such approaches often rely on the ability to optimally project a signal onto a small number of dictionary atoms. Such optimal, or even approximate, projections have been difficult to derive theoretically. Nonetheless, it has been observed experimentally that conventional CS approaches can be used for such projections, and still provide accurate signal recovery. In this letter, we summarize the empirical evidence and clearly demonstrate for what signal types certain CS methods may be used as approximate projections. In addition, we provide theoretical guarantees for such methods for certain sparse signal structures. Our theoretical results match those observed in experimental studies, and we thus establish both experimentally and theoretically that these CS methods can be used in this context. \end{abstract}