## The Annals of Statistics

### Accuracy guaranties for $\ell_{1}$ recovery of block-sparse signals

#### Abstract

We introduce a general framework to handle structured models (sparse and block-sparse with possibly overlapping blocks). We discuss new methods for their recovery from incomplete observation, corrupted with deterministic and stochastic noise, using block-$\ell_{1}$ regularization. While the current theory provides promising bounds for the recovery errors under a number of different, yet mostly hard to verify conditions, our emphasis is on verifiable conditions on the problem parameters (sensing matrix and the block structure) which guarantee accurate recovery. Verifiability of our conditions not only leads to efficiently computable bounds for the recovery error but also allows us to optimize these error bounds with respect to the method parameters, and therefore construct estimators with improved statistical properties. To justify our approach, we also provide an oracle inequality, which links the properties of the proposed recovery algorithms and the best estimation performance. Furthermore, utilizing these verifiable conditions, we develop a computationally cheap alternative to block-$\ell_{1}$ minimization, the non-Euclidean Block Matching Pursuit algorithm. We close by presenting a numerical study to investigate the effect of different block regularizations and demonstrate the performance of the proposed recoveries.

#### Article information

Source
Ann. Statist. Volume 40, Number 6 (2012), 3077-3107.

Dates
First available in Project Euclid: 22 February 2013

https://projecteuclid.org/euclid.aos/1361542075

Digital Object Identifier
doi:10.1214/12-AOS1057

Mathematical Reviews number (MathSciNet)
MR3097970

Zentralblatt MATH identifier
1296.62088

Subjects
Primary: 62G08: Nonparametric regression 62H12: Estimation
Secondary: 90C90: Applications of mathematical programming

#### Citation

Juditsky, Anatoli; Kılınç Karzan, Fatma; Nemirovski, Arkadi; Polyak, Boris. Accuracy guaranties for $\ell_{1}$ recovery of block-sparse signals. Ann. Statist. 40 (2012), no. 6, 3077--3107. doi:10.1214/12-AOS1057. https://projecteuclid.org/euclid.aos/1361542075

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#### Supplemental materials

• Supplementary material: Supplement to “Accuracy guaranties for $\ell_{1}$ recovery of block-sparse signals”. The proofs of the results stated in the paper and the derivations for Section 5.2 are provided in the supplementary article [18].