Frederic Matter (TU Darmstadt)
Sparse Recovery Under Side Constraints Using Null Space Properties
The problem of recovering a sparse vector via an underdetermined system of linear equations using a measurement matrix is one of the fundamental tasks in Compressed Sensing. In many applications, there is additional knowledge available, such as nonnegativity or integrality of the sparse vector, which can be exploited in the recovery problem. In order to characterize when recovery of sufficiently sparse vectors is possible, so-called null space properties (NSP) can be used. In this talk, we present a general framework for sparse recovery, which allows to incorporate additional knowledge in form of side constraints and propose a general NSP, which subsumes many specific settings already considered in the literature. This framework allows to analyze the influence of side constraints on the recovery process. For some explicit settings and side constraints, we derive specific NSPs and compare them. For the classical case of sparse vectors, we analyze the influence of nonnegativity by considering whether random measurement matrices satisfy the corresponding NSPs. Lastly, if time permits, we will shortly mention approaches to verify the NSP for a given measurement matrix.