On the Null Space Constant for ${\ell _p}$ Minimization

The literature on sparse recovery often adopts the lp “norm” ( p ∈ [0,1]) as the penalty to induce sparsity of the signal satisfying an underdetermined linear system. The performance of the corresponding lp 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 lp 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 lp minimization, mainly in the relationship between the the exponent p and the sparsity level k.