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

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, $$min\mathrm{\parallel }x{\mathrm{\parallel }}_{p,w}\text{ subject to }y=Ax,p=0,1,\text{ and }w\in [0,1],$$ uniqueness results are known when $w=0,1$. Here, $\mathrm{\parallel }x{\mathrm{\parallel }}_{p,w}=w\mathrm{\parallel }{x}_T{\mathrm{\parallel }}_p+\mathrm{\parallel }{x}_{T^c}{\mathrm{\parallel }}_p,p=0,1$ 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 $w\in [0,1]$.