Can we allow linear dependencies in the dictionary in the sparse synthesis framework?

TL;DR

This paper investigates whether linear dependencies in dictionaries can be allowed in the sparse synthesis framework, showing theoretically and empirically that such dependencies do not hinder and may even benefit signal recovery.

Contribution

It provides a theoretical proof that linear dependencies are permissible in the synthesis model and demonstrates empirical benefits for signal recovery.

Findings

Linear dependencies do not prevent signal recovery in the synthesis model.

Allowing dependencies can improve recovery performance.

Theoretical and empirical validation of dependencies in synthesis dictionaries.

Abstract

Signal recovery from a given set of linear measurements using a sparsity prior has been a major subject of research in recent years. In this model, the signal is assumed to have a sparse representation under a given dictionary. Most of the work dealing with this subject has focused on the reconstruction of the signal's representation as the means for recovering the signal itself. This approach forced the dictionary to be of low coherence and with no linear dependencies between its columns. Recently, a series of contributions that focus on signal recovery using the analysis model find that linear dependencies in the analysis dictionary are in fact permitted and beneficial. In this paper we show theoretically that the same holds also for signal recovery in the synthesis case for the l0- synthesis minimization problem. In addition, we demonstrate empirically the relevance of our conclusions for recovering the signal using an l1-relaxation.