Granulometric Smoothing on Manifolds
Abstract
Given a random sample from a density function supported on a manifold , a new method for the estimating highest density regions of the underlying population is introduced. The new proposal is based on the empirical version of the opening operator from mathematical morphology combined with a preliminary estimator of the density function. This results in an estimator that is easy-to-compute since it simply consists of a list of centers and a radius that are adequately selected from the data. The new estimator is shown to be consistent and its convergence rates in terms of the Hausdorff distance are provided. All consistency results are established uniformly on the level of the set and for any Riemannian manifold satisfying mild assumptions. The applicability of the procedure is shown by some illustrative examples.
Cite
@article{arxiv.2407.07559,
title = {Granulometric Smoothing on Manifolds},
author = {Diego Bolón and Rosa M. Crujeiras and Alberto Rodríguez-Casal},
journal= {arXiv preprint arXiv:2407.07559},
year = {2026}
}
Comments
65 pages (a main paper of 28 pages and several appendices)