The Sparsity of Underdetermined Linear System via lp Minimization for 0<1

• Published in: Mathematical problems in engineering Vol. 2015
• Main Authors: Li, Haiyang, Peng, Jigen, Yue, Shigang
• Format: Journal Article
• Language: English
• Published: 01.01.2015
• Subjects: Constants; Linear systems; Mathematical analysis; Minimization; Optimization; Permutations; Platinum; Representations
• ISSN: 1024-123X; 1563-5147
• DOI: 10.1155/2015/584712

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 sub(lp) minimization for 0<1. We show, for a given underdetermined linear system of equations sub(Amn) X=b, that although it is not certain that the problem ( sub(Pp) ) (i.e., sub(minX) super(Xpp) subject to AX=b, where 0<1) generates sparser solutions as the value of p decreases and especially the problem ( sub(Pp) ) generates sparser solutions than the problem ( sub(P1) ) (i.e., sub(minX) sub(X1) subject to AX=b), there exists a sparse constant gamma (A,b)>0 such that the following conclusions hold when p< gamma (A,b): (1) the problem ( sub(Pp) ) generates sparser solution as the value of p decreases; (2) the sparsest optimal solution to the problem ( sub(Pp) ) is unique under the sense of absolute value permutation; (3) let sub(X1) and sub(X2) be the sparsest optimal solution to the problems ( sub(Pp1)) and ( sub(Pp2)) ( sub(p1) < sub(p2) ), respectively, and let sub(X1) not be the absolute value permutation of sub(X2) . Then there exist sub(t1) , sub(t2) [is an element of][ sub(p1) , sub(p2) ] such that sub(X1) is the sparsest optimal solution to the problem ( sub(Pt) ) ([forall]t[is an element of][ sub(p1) , sub(t1) ]) and sub(X2) is the sparsest optimal solution to the problem ( sub(Pt) ) ([forall]t[is an element of]( sub(t2) , sub(p2) ]).