### The Sparsity of Underdetermined Linear System vialpMinimization for0<p<1

Haiyang Li, Jigen Peng, Shigang Yue
2015 Mathematical Problems in Engineering
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 vialpminimization for0<p<1. We show, for a given underdetermined linear system of equationsAm×nX=b, that although it is not certain that the problem(Pp)(i.e.,minXXppsubject toAX=b, where0<p<1) generates sparser solutions as the value ofpdecreases and especially the
more » ... nd especially the problem(Pp)generates sparser solutions than the problem(P1)(i.e.,minXX1subject toAX=b), there exists a sparse constantγ(A,b)>0such that the following conclusions hold whenp<γ(A,b):(1)the problem(Pp)generates sparser solution as the value ofpdecreases;(2)the sparsest optimal solution to the problem(Pp)is unique under the sense of absolute value permutation;(3)letX1andX2be the sparsest optimal solution to the problems(Pp1)and(Pp2) (p1<p2), respectively, and letX1not be the absolute value permutation ofX2. Then there existt1,t2∈[p1,p2]such thatX1is the sparsest optimal solution to the problem(Pt) (∀t∈[p1,t1])andX2is the sparsest optimal solution to the problem(Pt) (∀t∈(t2,p2]).