On the Null Space Constant for $l_p$ Minimization

The literature on sparse recovery often adopts the $l_p$ "norm" $(p\in[0,1])$ as the penalty to induce sparsity of the signal satisfying an underdetermined linear system. The performance of the corresponding $l_p$ 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 $l_p$ 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 ${\bf A}$ is randomly generated. Finally, we show how these properties can help in demonstrating the performance of $l_p$ minimization, mainly in the relationship between the the exponent $p$ and the sparsity level $k$.