Multiple Choice Hard Thresholding Pursuit (MCHTP) for Simultaneous Sparse Recovery and Sparsity Order Estimation

We address the problem of sparse recovery using greedy compressed sensing recovery algorithms, without explicit knowledge of the sparsity. Estimating the sparsity order is a crucial problem in many practical scenarios, e.g., wireless communications, where exact value of the sparsity order of the unknown channel may be unavailable a priori. In this paper we have proposed a new greedy algorithm, referred to as the Multiple Choice Hard Thresholding Pursuit (MCHTP), which modifies the popular hard thresholding pursuit (HTP) suitably to iteratively recover the unknown sparse vector along with the sparsity order of the unknown vector. We provide provable performance guarantees which ensures that MCHTP can estimate the sparsity order exactly, along with recovering the unknown sparse vector exactly with noiseless measurements. The simulation results corroborate the theoretical findings, demonstrating that even without exact sparsity knowledge, with only the knowledge of a loose upper bound of the sparsity, MCHTP exhibits outstanding recovery performance, which is almost identical to that of the conventional HTP with exact sparsity knowledge. Furthermore, simulation results demonstrate much lower computational complexity of MCHTP compared to other state-of-the-art techniques like MSP.