### Abstract

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 ^{2}I), 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.

Original language | English (US) |
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Title of host publication | Advances in Neural Information Processing Systems 23: 24th Annual Conference on Neural Information Processing Systems 2010, NIPS 2010 |

State | Published - 2010 |

Event | 24th Annual Conference on Neural Information Processing Systems 2010, NIPS 2010 - Vancouver, BC, Canada Duration: Dec 6 2010 → Dec 9 2010 |

### Other

Other | 24th Annual Conference on Neural Information Processing Systems 2010, NIPS 2010 |
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Country | Canada |

City | Vancouver, BC |

Period | 12/6/10 → 12/9/10 |

### Fingerprint

### ASJC Scopus subject areas

- Information Systems

### Cite this

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

**Multi-stage Dantzig selector.** / Liu, Ji; Wonka, Peter; Ye, Jieping.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Advances in Neural Information Processing Systems 23: 24th Annual Conference on Neural Information Processing Systems 2010, NIPS 2010.*24th Annual Conference on Neural Information Processing Systems 2010, NIPS 2010, Vancouver, BC, Canada, 12/6/10.

}

TY - GEN

T1 - Multi-stage Dantzig selector

AU - Liu, Ji

AU - Wonka, Peter

AU - Ye, Jieping

PY - 2010

Y1 - 2010

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

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

M3 - Conference contribution

AN - SCOPUS:84860621071

SN - 9781617823800

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

ER -