On NP-Hardness of $L_1/L_2$ Minimization and Bound Theory of Nonzero Entries in Solutions

TL;DR

This paper proves that finding the global minimum of the L1/L2 minimization problem is strongly NP-hard and provides bounds on the solutions' properties, extending the analysis to other Lp/Lq models.

Contribution

It establishes the NP-hardness of L1/L2 minimization and derives bounds on local minimizers and nonzero entries, advancing theoretical understanding of these models.

Findings

Global minimum finding is strongly NP-hard.

Uniform upper bounds on L2 norm of local minimizers.

Bounds on nonzero entries in solutions.

Abstract

The \(L_1/L_2\) norm ratio has gained significant attention as a measure of sparsity due to three merits: sharper approximation to the \(L_0\) norm compared to the \(L_1\) norm, being parameter-free and scale-invariant, and exceptional performance with highly coherent matrices. These properties have led to its successful application across a wide range of fields. While several efficient algorithms have been proposed to compute stationary points for \(L_1/L_2\) minimization problems, their computational complexity has remained open. In this paper, we prove that finding the global minimum of both constrained and unconstrained \(L_1/L_2\) models is strongly NP-hard. In addition, we establish uniform upper bounds on the \(L_2\) norm for any local minimizer of both constrained and unconstrained \(L_1/L_2\) minimization models. We also derive upper and lower bounds on the magnitudes of the nonzero entries in any local minimizer of the unconstrained model, aiding in classifying nonzero entries. Finally, we extend our analysis to demonstrate that the constrained and unconstrained \(L_p/L_q\) (\(0 < p \leq 1, 1 < q < +\infty\)) models are also strongly NP-hard.