TY - GEN

T1 - Multi-stage Dantzig selector

AU - Liu, Ji

AU - Wonka, Peter

AU - Ye, Jieping

PY - 2010/12/1

Y1 - 2010/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84860621071&partnerID=8YFLogxK

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M3 - Conference contribution

AN - SCOPUS:84860621071

SN - 9781617823800

T3 - Advances in Neural Information Processing Systems 23: 24th Annual Conference on Neural Information Processing Systems 2010, NIPS 2010

BT - Advances in Neural Information Processing Systems 23

T2 - 24th Annual Conference on Neural Information Processing Systems 2010, NIPS 2010

Y2 - 6 December 2010 through 9 December 2010

ER -