Reconstruction of atomic measures from their halfspace depth

The halfspace depth can be seen as a mapping that to a finite Borel measure μ on the Euclidean space Rd assigns its depth, being a function Rd→[0,∞):x↦Dx;μ. The depth of μ quantifies how much centrally positioned a point x is with respect to μ. This function is intended to serve as generalization of the quantile function to multivariate spaces. We consider the problem of finding the inverse mapping to the halfspace depth: knowing only the function x↦Dx;μ, our objective is to reconstruct the measure μ. We focus on μ atomic with finitely many atoms, and present a simple method for the reconstruction of the position and the weights of all atoms of μ, from its depth only. As a consequence, (i) we recover generalizations of several related results known from the literature, with substantially simplified proofs, and (ii) design a novel reconstruction procedure that is numerically more stable, and considerably faster than the known algorithms. Our analysis presents a comprehensive treatment of the halfspace depth of those measures whose depths attain finitely many different values.