Multi-stage Dantzig selector

We consider the following sparse signal recovery (or feature selection) problem: given a design matrix X ∈ ℝ ^n×m (m ≫ n) and a noisy observation vector y ∈ ℝ ⁿ satisfying y = Xβ* +ε where ε is the noise vector following a Gaussian distribution N(0; σ ²I), 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 l _p norm (p ≥ 1) is bounded as ∥ β̂- β* _p≤ (C(s - N) ^1/p √ logm + Δ)σ where C is a constant, s is the number of nonzero entries in β* Δ 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 √ logm). The proposed method improves the estimation bound of the standard Dantzig selector approximately from Cs ^1/p √ logmσ to C(s-N) ^1/p √ logmσ 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.