Weighted LASSO for Sparse Recovery With Statistical Prior Support Information

Abstract

Compressive sensing is used to recover a sparse signal from linear measurements. Without any prior support information (PSI), least absolute shrinkage and selection operator (LASSO) is a useful method for sparse recovery. In some settings, a statistical prior about the support of the sparse signal may be provided. It is critical to optimally incorporate such statistical PSI to enhance the recovery performance. We propose a weighted LASSO algorithm to fully exploit the statistical PSI and optimize the recovery performance. In the proposed algorithm, we exploit the most general statistical PSI model, a multilevel PSI, and incorporate it into the LASSO using a weighted $l_{1}$ norm penalty function. An optimal weight policy is proposed to minimize the asymptotic normalized squared error (aNSE). We also derive the closed-form accurate expression for the minimum aNSE and characterize the minimum number of measurements required to achieve stable recovery. Based on this, we discuss the impact of PSI quality on the aNSE performance of the proposed algorithm. Both theoretical analysis and simulations show the performance advantages of our proposed solution over various baselines.