The sparsity of underdetermined linear system via lp minimization for 0 < p < 1

Li, Haiyang and Peng, Jigen and Yue, Shigang (2015) The sparsity of underdetermined linear system via lp minimization for 0 < p < 1. Mathematical Problems in Engineering, 2015 . ISSN 1024-123X

Full content URL: http://www.hindawi.com/journals/mpe/2015/584712/

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The sparsity of underdetermined linear system via lp minimization for 0 < p < 1
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Abstract

The sparsity problems have attracted a great deal of attention in recent years, which aim to find the sparsest solution of a representation or an equation. In the paper, we mainly study the sparsity of underdetermined linear system via lp minimization for 0<p<1. We show, for a given underdetermined linear system of equations pm×np = p, that although it is not certain that the problem (pp) (i.e., minlx||X||plp subject to pp = b, where 0<p<1 ) generates sparser solutions as the value of p decreases and especially the problem (plp) generates sparser solutions than the problem (p1) (i.e., minlx||X||1 subject to AX = b ), there exists a sparse constant γ(A, p) > 0 such that the following conclusions hold when p < γ(A, b): (1) the problem (pp) generates sparser solution as the value of p decreases; (2) the sparsest optimal solution to the problem (pp) is unique under the sense of absolute value permutation; (3) let X1 and X2 be the sparsest optimal solution to the problems (pp1) and (pp2) , respectively, and let X1 not be the absolute value permutation of X2. Then there exist t1,t2 ε [p1,p2] such that X1 is the sparsest optimal solution to the problem (pt) (∀t ε [p1, t1]) and X2 is the sparsest optimal solution to the problem (pt) (∀t ε (t2, p2]).

Additional Information:Article ID 584712, 6 pages
Keywords:Optimal systems, Absolute values, Optimal solutions, Sparsity problems, Underdetermined linear systems, Linear systems, JCOpen
Subjects:G Mathematical and Computer Sciences > G100 Mathematics
G Mathematical and Computer Sciences > G160 Engineering/Industrial Mathematics
Divisions:College of Science > School of Computer Science
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ID Code:17577
Deposited On:12 Jun 2015 09:57

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