Sufficient conditions for the uniqueness of solution of the weighted norm minimization problem

TL;DR

This paper establishes conditions under which the weighted norm minimization problem in compressed sensing has a unique solution for any weight in [0,1], extending known results from the extreme weights 0 and 1.

Contribution

It provides new sufficient conditions for the uniqueness of solutions in weighted norm minimization with arbitrary weights, generalizing previous results.

Findings

Derived conditions for solution uniqueness for all weights in [0,1]

Extended known results from weights 0 and 1 to the entire interval

Enhances understanding of support constrained compressed sensing

Abstract

Prior support constrained compressed sensing, achieved via the weighted norm minimization, has of late become popular due to its potential for applications. For the weighted norm minimization problem, $$ min \|x\|_{p,w} \text{ subject to } y=Ax, \; p=0,1, \text{ and } w \in [0,1], $$ uniqueness results are known when $w=0,1$. Here, $\|x\|_{p,w}=w\|x_T\|_p+\|x_{T^c}\|_p, \; p=0,1$ with $T$ representing the partial support information. The work reported in this paper presents the conditions that ensure the uniqueness of the solution of this problem for general $w \in [0,1]$.