Greedy Signal Space Methods for incoherence and beyond

TL;DR

This paper analyzes greedy signal space methods for signal recovery in overcomplete dictionaries, providing new theoretical insights that extend their applicability to incoherent and structured dictionaries, surpassing previous limitations.

Contribution

It offers an alternative analysis of greedy methods that relax projection accuracy requirements, applicable to incoherent and structured dictionaries, improving theoretical understanding.

Findings

Analysis aligns with traditional CoSa results

Applicable to incoherent and structured dictionaries

Improves theoretical bounds for greedy methods

Abstract

Compressive sampling (CoSa) has provided many methods for signal recovery of signals compressible with respect to an orthonormal basis. However, modern applications have sparked the emergence of approaches for signals not sparse in an orthonormal basis but in some arbitrary, perhaps highly overcomplete, dictionary. Recently, several "signal-space" greedy methods have been proposed to address signal recovery in this setting. However, such methods inherently rely on the existence of fast and accurate projections which allow one to identify the most relevant atoms in a dictionary for any given signal, up to a very strict accuracy. When the dictionary is highly overcomplete, no such projections are currently known; the requirements on such projections do not even hold for incoherent or well-behaved dictionaries. In this work, we provide an alternate analysis for signal space greedy methods which enforce assumptions on these projections which hold in several settings including those when the dictionary is incoherent or structurally coherent. These results align more closely with traditional results in the standard CoSa literature and improve upon previous work in the signal space setting.