DocumentCode :
1423708
Title :
\\ell _{2}/\\ell _{1} -Optimization in Block-Sparse Compressed Sensing and Its Strong Thresholds
Author :
Stojnic, Mihailo
Author_Institution :
Sch. of Ind. Eng., Purdue Univ., West Lafayette, IN, USA
Volume :
4
Issue :
2
fYear :
2010
fDate :
4/1/2010 12:00:00 AM
Firstpage :
350
Lastpage :
357
Abstract :
It has been known for a while that l1-norm relaxation can in certain cases solve an under-determined system of linear equations. Recently, E. Candes ("Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information," IEEE Trans. Information Theory, vol. 52, no. 12, pp. 489-509, Dec. 2006) and D. Donoho ("High-dimensional centrally symmetric polytopes with neighborlines proportional to dimension," Disc. Comput. Geometry, vol. 35, no. 4, pp. 617-652, 2006) proved (in a large dimensional and statistical context) that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of nonzero elements of the unknown vector) also proportional to the length of the unknown vector such that l1-norm relaxation succeeds in solving the system. In this paper, in a large dimensional and statistical context, we determine sharp lower bounds on the values of allowable sparsity for any given number (proportional to the length of the unknown vector) of equations for the case of the so-called block-sparse unknown vectors considered in "On the reconstruction of block-sparse signals with an optimal number of measurements," (M. Stojnic et al., IEEE Trans, Signal Processing, submitted for publication.
Keywords :
block codes; data compression; optimisation; statistical analysis; block-sparse compressed sensing; block-sparse signal; block-sparse unknown vector; l1-norm relaxation; l1-optimization; l2-optimization; sharp lower bounds; statistical context; Compressed sensing; Equations; Frequency; Geometry; Information theory; Length measurement; Signal processing; Signal reconstruction; Terminology; Uncertainty; $ell_{2}/ell_{1}$ -optimization; Block-sparse; compressed sensing;
fLanguage :
English
Journal_Title :
Selected Topics in Signal Processing, IEEE Journal of
Publisher :
ieee
ISSN :
1932-4553
Type :
jour
DOI :
10.1109/JSTSP.2009.2039172
Filename :
5419037
Link To Document :
بازگشت