A Multi-Stage Framework for Dantzig Selector and LASSO

We consider the following sparse signal recovery (or feature selection) problem: given a design matrix X ∈ Rn×m (m ≫ n) and a noisy observation vector y ∈ Rn satisfying y = Xβ∗ + ε where ε is the noise vector following a Gaussian distribution N(0,σ2I), how to recover the signal (or parameter vector) β∗ when the signal is sparse? The Dantzig selector has been proposed for sparse signal recovery with strong theoretical guarantees. In this paper, we propose a multi-stage Dantzig selector method, which iteratively refines the target signal β∗. We show that if X obeys a certain condition, then with a large probability the difference between the solution β̂ estimated by the proposed method and the true solution β∗ measured in terms of the lp norm (p ≥ 1) is bounded as ‖β̂−β‖p ≤ (