### Abstract

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 ^{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*, 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.

Original language | English (US) |
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Pages (from-to) | 1189-1219 |

Number of pages | 31 |

Journal | Journal of Machine Learning Research |

Volume | 13 |

State | Published - Apr 1 2012 |

### Keywords

- Dantzig selector
- LASSO
- Multi-stage
- Sparse signal recovery

### ASJC Scopus subject areas

- Software
- Control and Systems Engineering
- Statistics and Probability
- Artificial Intelligence

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## Cite this

*Journal of Machine Learning Research*,

*13*, 1189-1219.