Cross Low-Dimension Pursuit for Sparse Signal Recovery from Incomplete Measurements Based on Permuted Block Diagonal Matrix

Zaixing HE
Takahiro OGAWA

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E94-A    No.9    pp.1793-1803
Publication Date: 2011/09/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E94.A.1793
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Digital Signal Processing
sparse recovery,  sparsest solution,  compressed sensing,  permuted block diagonal matrix,  greedy algorithms,  orthogonal matching pursuit,  1-norm minimization,  basis pursuit,  

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In this paper, a novel algorithm, Cross Low-dimension Pursuit, based on a new structured sparse matrix, Permuted Block Diagonal (PBD) matrix, is proposed in order to recover sparse signals from incomplete linear measurements. The main idea of the proposed method is using the PBD matrix to convert a high-dimension sparse recovery problem into two (or more) groups of highly low-dimension problems and crossly recover the entries of the original signal from them in an iterative way. By sampling a sufficiently sparse signal with a PBD matrix, the proposed algorithm can recover it efficiently. It has the following advantages over conventional algorithms: (1) low complexity, i.e., the algorithm has linear complexity, which is much lower than that of existing algorithms including greedy algorithms such as Orthogonal Matching Pursuit and (2) high recovery ability, i.e., the proposed algorithm can recover much less sparse signals than even 1-norm minimization algorithms. Moreover, we demonstrate both theoretically and empirically that the proposed algorithm can reliably recover a sparse signal from highly incomplete measurements.