On uniqueness guarantees of solution in convex regularized linear inverse problems

This paper is concerned with uniqueness guarantees of solution in a class of convex regularized linear inverse problems. The existence of a non-degenerate dual certificate was previously known to be sufficient for a generalized version of the null space property to hold, and hence ensures uniqueness of solution. In this paper, we prove that they are actually equivalent under a restricted injectivity condition. By constructing counter-examples, we show that they are sufficient but not necessary for the uniqueness of solution in general cases.