  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}
