The Sparsity of Underdetermined Linear System via lp Minimization for 0<p<1 - Li, Haiyang; Yue, Shigang; Peng, Jigen (Mathematical problems in engineering, 2015)
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 Am×nX=b, that although it is not certain that the problem (Pp) (i.e., minXXpp subject to AX=b, where 0<p<1) generates sparser solutions as the value of p decreases and especially the problem (Pp) generates sparser solutions than the problem (P1) (i.e., minXX1 subject to AX=b), there exists a sparse constant γ(A,b)>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) (p1<p2), respectively, and let X1 not be the absolute value permutation of X2. Then there exist t1,t2∈[p1,p2] such that X1 is the sparsest optimal solution to the problem (Pt) (∀t∈[p1,t1]) and X2 is the sparsest optimal solution to the problem (Pt) (∀t∈(t2,p2]).