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Multi-Stage Dantzig Selector

By Ji Liu, Peter Wonka and Jieping Ye


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 ∗ + ɛ where ɛ 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 ˆβ estimated by the proposed method and the true solution β ∗ measured in terms of the lp norm (p ≥ 1) is bounded as ‖ ˆ β − 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 1/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. 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.

Year: 2012
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