ACCURACY GUARANTIES FOR l1 RECOVERY OF BLOCK-SPARSE SIGNALS

• 成果类型: Article
• 署名作者: Juditsky, Anatoli; Karzan, Fatma Kilinc; Nemirovski, Arkadi; Polyak, Boris
• 署名单位: Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Carnegie Mellon University; University System of Georgia; Georgia Institute of Technology; V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences; Russian Academy of Sciences
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/12-AOS1057
• 发表日期: 2012
• 页码: 3077-3107
• 关键词: group lasso; DANTZIG SELECTOR; REPRESENTATIONS; UNION

摘要

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-l(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-l(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.