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

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