On recovery of sparse signals with block structures

It has been widely recognized that structure information helps in sparse signal recovery. In this paper, a general form of block structure is considered, which is often referred to hierarchically sparse model. It is assumed that the unknown sparse signal can be divided into blocks, and a block contains either all zero components or a fraction of nonzero components. This model sits between the standard sparse model (without block structure) and the strict block sparse model (all entries in nonzero blocks are nonzero). The focus of this paper is to analyze the convex optimization approach to recover hierarchically sparse signals. The technique we employed is based on the approximated message passing framework and the associated state evolution. The minimum number of measurements required for exact recovery, also known as phase transition (PT), has been quantified in an asymptotic region. We show that the PT depends on two parameters: the fraction of nonzero components in nonzero blocks, and the uniformity of the magnitudes of nonzero components. Based on the PT analysis, we characterize the regions at which the convex optimization methods designed for the standard, hierarchically, and block sparse models are optimal respectively.