Open Access
December 2012 Accuracy guaranties for $\ell_{1}$ recovery of block-sparse signals
Anatoli Juditsky, Fatma Kılınç Karzan, Arkadi Nemirovski, Boris Polyak
Ann. Statist. 40(6): 3077-3107 (December 2012). DOI: 10.1214/12-AOS1057


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.


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Anatoli Juditsky. Fatma Kılınç Karzan. Arkadi Nemirovski. Boris Polyak. "Accuracy guaranties for $\ell_{1}$ recovery of block-sparse signals." Ann. Statist. 40 (6) 3077 - 3107, December 2012.


Published: December 2012
First available in Project Euclid: 22 February 2013

zbMATH: 1296.62088
MathSciNet: MR3097970
Digital Object Identifier: 10.1214/12-AOS1057

Primary: 62G08 , 62H12
Secondary: 90C90

Keywords: nonparametric estimation by convex optimization , Oracle inequalities , sparse recovery

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 6 • December 2012
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