Sparsity according to Prony, average performance analysis

Finding the sparse representation of a signal in an overcomplete dictionary has attracted a lot of attention over the past years. Traditional approaches such as Basis Pursuit are based on relaxing a nonconvex `0-minimization problem [1]–[3]. In [4], a new polynomial complexity algorithm, ProSparse, is presented. ProSparse solves the sparse representation problem when the dictionary is the union of Fourier and canonical bases and can be extended to other relevant pairs of bases or frames. Here, we present a probabilistic average-case analysis that characterizes a sharp phase transition behaviour of the algorithm. We also present an extension of the algorithm for the noisy scenario. This proposed extension outperforms the Basis Pursuit Denoise algorithm in support retrieval in a number of scenarios.