The Sparsity of Underdetermined Linear System via lp Minimization for 0

Abstract EN

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

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 00 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

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]).