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 ε ℝ ^n×m (m n) and a noisy observation vector y ε 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*, the risk of the oracle estimator 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, where the oracle solution is the projection of the observation vector y onto true features. 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. Finally, we extend this multi-stage procedure to the LASSO case.