  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.
