NP/CMP equivalence: a phenomenon hidden among sparsity models l_{0} minimization and l_{p} minimization for information processing

In this paper, we have proved that in every underdetermined linear system Ax = b, there corresponds a constant p*(A, b) > 0 such that every solution to the l p-norm minimization problem also solves the l0-norm minimization problem whenever 0 <; p <; p*(A, b). This phenomenon is named NP/CMP equivalence

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

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 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])