A Multi-Stage Framework for Dantzig Selector and LASSO
Ji Liu, Peter Wonka, Jieping Ye.
Year: 2012, Volume: 13, Issue: 41, Pages: 1189−1219
Abstract
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≤ (C(s-N)1/p√log m+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√log m). The proposed method improves the estimation bound of the standard Dantzig selector approximately from Cs1/p√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, 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.