On the Null Space Constant for $l_p$ Minimization

TL;DR

This paper investigates the properties of the null space constant in $l_p$ minimization, revealing its monotonicity and continuity, which aids in understanding sparse recovery performance.

Contribution

It establishes the strict increase of the null space constant with sparsity level and exponent, providing insights into $l_p$ minimization performance.

Findings

Null space constant increases with sparsity level $k$

Null space constant is continuous in $p$

Constant increases with $p$ when sensing matrix is random

Abstract

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 $k$.