Denoising and Interior Detection Problems

TL;DR

This paper investigates methods to determine whether a manifold's interior exists based on point samples, addressing dependence and noise issues, and proposing techniques for accurate interior detection and noise filtering.

Contribution

It introduces a novel analysis of interior detection tests under dependent and noisy sampling conditions, with convergence guarantees and noise identification strategies.

Findings

Asymptotic properties of interior detection tests are characterized.

A methodology for distinguishing true manifold points from noisy observations is developed.

Convergence properties of the proposed methods are established.

Abstract

Let $\mathcal{M}$ be a compact manifold of $\mathbb{R}^d$. The goal of this paper is to decide, based on a sample of points, whether the interior of $\mathcal{M}$ is empty or not. We divide this work in two main parts. Firstly, under a dependent sample which may or may not contain some noise within, we characterize asymptotic properties of an interior detection test based on a suitable control of the dependence. Afterwards, we drop the dependence and consider a model where the points sampled from the manifold are mixed with some points sampled from a different measure (noisy observations). We study the behaviour with respect to the amount of noisy observations, introducing a methodology to identify true manifold points, characterizing convergence properties.