Multi-Stage Dantzig Selector

We consider the following sparse signal recovery (or feature selection) problem: given a design matrix X ∈ Rn×m (m À n) and a noisy observation vector y ∈ R satisfying y = Xβ∗ + 2 where 2 is the noise vector following a Gaussian distribution N(0, σI), 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 ≤ ( C(s−N)1/p √ log m + ∆ ) σ, where C is a constant, s is the number of nonzero entries in β∗, ∆ is independent of m and is much smaller than the first term, and N is the number of entries of β∗ larger than a certain value in the order of O(σ√log m). The proposed method improves the estimation bound of the standard Dantzig selector approximately from Cs √ log mσ to C(s−N)1/p√log mσ where the value N depends on the number of large entries in β∗. When N = s, the proposed algorithm achieves the oracle solution with a high probability. In addition, with a large probability, the proposed method can select the same number of correct features under a milder condition than the Dantzig selector.