Tight Bounds for the Learning of Homotopy à la Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Preprints, Working Papers, ... Year : 2024

Tight Bounds for the Learning of Homotopy à la Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds

Abstract

In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on the learning of the homotopy type from a sample of an underlying space. In their work, Niyogi, Smale, and Weinberger studied samples of C2 manifolds with positive reach embedded in ℝd. We extend their results in the following ways: In the first part of our paper we consider both manifolds of positive reach -- a more general setting than C2 manifolds -- and sets of positive reach embedded in ℝd. The sample P of such a set S does not have to lie directly on it. Instead, we assume that the two one-sided Hausdorff distances -- ε and δ -- between P and S are bounded. We provide explicit bounds in terms of ε and δ, that guarantee that there exists a parameter r such that the union of balls of radius r centred at the sample P deformation-retracts to S. In the second part of our paper we study homotopy learning in a significantly more general setting -- we investigate sets of positive reach and submanifolds of positive reach embedded in a Riemannian manifold with bounded sectional curvature. To this end we introduce a new version of the reach in the Riemannian setting inspired by the cut locus. Yet again, we provide tight bounds on ε and δ for both cases (submanifolds as well as sets of positive reach), exhibiting the tightness by an explicit construction.
Fichier principal
Vignette du fichier
NSW_arXiv.pdf (31.48 Mo) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-03721463 , version 1 (12-07-2022)
hal-03721463 , version 2 (07-03-2024)

Identifiers

Cite

Dominique Attali, Hana Dal Poz Kouřimská, Christopher Fillmore, Ishika Ghosh, André Lieutier, et al.. Tight Bounds for the Learning of Homotopy à la Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds. 2024. ⟨hal-03721463v2⟩
149 View
16 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More