Sparse Decomposition over non-full-rank dictionaries

Sparse Decomposition (SD) of a signal on an overcomplete dictionary has recently attracted a lot of interest in signal processing and statistics, because of its potential application in many different areas including Compressive Sensing (CS). However, in the current literature, the dictionary matrix has generally been assumed to be of full-rank. In this paper, we consider non-full-rank dictionaries (which are not even necessarily overcomplete), and extend the definition of SD over these dictionaries. Moreover, we present an approach which enables to use previously developed SD algorithms for this non-full-rank case. Besides this general approach, for the special case of the Smoothed ℓ0 (SL0) algorithm, we show that a slight modification of it covers automatically non-full-rank dictionaries.