Multiple Choice Hard Thresholding Pursuit (MCHTP) for simultaneous sparse recovery and sparsity order estimation

Abstract

In this work, we propose a new greedy algorithm, referred to as the Multiple Choice Hard Thresholding Pursuit (MCHTP) which recovers a sparse vector from compressed linear measurements without explicit knowledge of its sparsity. MCHTP achieves this by suitably combining the steps of hard thresholding pursuit (HTP) with a novel sparsity selection criterion to iteratively select one among two possible sparsity values and subsequently recovering the unknown sparse vector along with its unknown sparsity order. We provide provable theoretical guarantees which ensure that MCHTP can estimate the sparsity order as well as the unknown sparse vector exactly from linear measurements. We also propose an extension of MCHTP, referred to as MCHTP−d, which can provide finer control on the estimated sparsity by tuning the parameter d. The simulation results corroborate the theoretical findings demonstrating the superior performance of MCHTP and MCHTP−d compared to state-of-the-art techniques in terms of sparse recovery and sparsity order estimation.