On the Null Space Constant for ℓp Minimization

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.