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Mathematical Problems in Engineering
Volume 2014, Article ID 358742, 23 pages
Research Article

A Greedy Multistage Convex Relaxation Algorithm Applied to Structured Group Sparse Reconstruction Problems Based on Iterative Support Detection

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

Received 27 April 2014; Revised 27 August 2014; Accepted 29 August 2014; Published 21 October 2014

Academic Editor: Yi-Kuei Lin

Copyright © 2014 Liangtian He and Yilun Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We propose a new effective algorithm for recovering a group sparse signal from very limited observations or measured data. As we know that a better reconstruction quality can be achieved when encoding more structural information besides sparsity, the commonly employed -regularization incorporating the prior grouping information has a better performance than the plain -regularized models as expected. In this paper we make a further use of the prior grouping information as well as possibly other prior information by considering a weighted model. Specifically, we propose a multistage convex relaxation procedure to alternatively estimate weights and solve the resulted weighted problem. The procedure of estimating weights makes better use of the prior grouping information and is implemented based on the iterative support detection (Wang and Yin, 2010). Comprehensive numerical experiments show that our approach brings significant recovery enhancements compared with the plain model, solved via the alternating direction method (ADM) (Deng et al., 2013), either in noiseless or in noisy environments.