On the Null Space Constant for <formula formulatype="inline"><tex Notation="TeX">${\ell _p}$</tex></formula> Minimization

Abstract

The literature on sparse recovery often adopts the p "norm" (p ∈ [0, 1]) as the penalty to induce sparsity of the signal satisfying an underdetermined linear system. The performance of the corresponding p minimization problem can be characterized by its null space constant. In spite of the NP-hardness of computing the constant, its properties can still help in illustrating the performance of p minimization. In this letter, we show the strict increase of the null space constant in the sparsity level k and its continuity in the exponent p. We also indicate that the constant is strictly increasing in p with probability 1 when the sensing matrix A is randomly generated. Finally, we show how these properties can help in demonstrating the performance of p minimization, mainly in the relationship between the the exponent p and the sparsity level k.

Analysis of the equivalence relationship between l 0 $l_{0}$ -minimization and l p

“…Combining the definition of NSC and the results in [23] and [22], we can derive the following corollaries.…”

“…Although it is NP-hard to find the global optimal solution of -minimization, a local minimizer can be done in polynomial time [24]. Chen [22] proved that is a necessary and sufficient condition for the global optimality of -minimization. Therefore, it is confident that we can find the sparse solution with -minimization with as long as we start with a good initialization.…”

“…Chen and Gu [22] showed some important properties of . It is shown that is a continuous function in when , where is the smallest number of columns from A that are linearly dependent.…”