Blind cluster structured sparse signal recovery: A nonconvex approach

• We focus on recovering cluster structured sparse signals from few acquisitions without any information on the signal structure.
• We consider a compressed sensing approach with a nonconvex nonseparable neighborhood based sparsity inducing function.
• We solve the corresponding constrained nonconvex minimization problem by integrating an IRl1 scheme into the penalization approach.
• The convergence of the iterative algorithm to a local minimum of the original problem is guaranteed.
• The local nature of the approach allows the algorithm to learn the unknown signal structure during the reconstruction process.

We consider the problem of recovering a sparse signal when its nonzero coefficients tend to cluster into blocks, whose number, dimension and position are unknown. We refer to this problem as blind cluster structured sparse recovery. For its solution, differently from the existing methods that consider the problem in a statistical context, we propose a deterministic neighborhood based approach characterized by the use both of a nonconvex, nonseparable sparsity inducing function and of a penalized version of the iterative ℓ1 reweighted method. Despite the high nonconvexity of the approach, a suitable integration of these building elements led to the development of MB-NFCS (Model Based Nonlinear Filtering for Compressed Sensing), an iterative fast, self-adaptive, and efficient algorithm that, without requiring any information on the sparsity pattern, adjusts at each iteration the action of the sparsity inducing function in order to strongly encourage the emerging cluster structure. The effectiveness of the proposed approach is demonstrated by a large set of numerical experiments that show the superior performance of MB-NFCS to the state-of-the-art algorithms.