Noisy Sparse Recovery Based on Parameterized Quadratic Programming by Thresholding

Parameterized quadratic programming (Lasso) is a powerful tool for the recovery of sparse signals based on underdetermined observations contaminated by noise. In this paper, we study the problem of simultaneous sparsity pattern recovery and approximation recovery based on the Lasso. An extended Lasso method is proposed with the following main contributions: (1) we analyze the recovery accuracy of Lasso under the condition of guaranteeing the recovery of nonzero entries positions. Specifically, an upper bound of the tuning parameter h of Lasso is derived. If h exceeds this bound, the recovery error will increase with h; (2) an extended Lasso algorithm is developed by choosing the tuning parameter according to the bound and at the same time deriving a threshold to recover zero entries from the output of the Lasso. The simulation results validate that our method produces higher probability of sparsity pattern recovery and better approximation recovery compared to two state-of-the-art Lasso methods.