Tight Bounds for the Learning of Homotopy à la Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds - Archive ouverte HAL
Communication Dans Un Congrès Année : 2024

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

Résumé

In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on thelearning 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 extendtheir results in the following ways:- As the ambient space we consider both Rd and Riemannian manifolds with lower boundedsectional curvature.- In both types of ambient spaces, we study sets of positive reach – a significantly more generalsetting than C2 manifolds – as well as general manifolds of positive reach.- The sample P of a set (or a manifold) S of positive reach may be noisy. We work with twoone-sided Hausdorff distances – ε and δ – between P and S. We provide tight bounds in termsof ε and δ, that guarantee that there exists a parameter r such that the union of balls of radiusr centred at the sample P deformation-retracts to S. We exhibit their tightness by an explicitconstruction.We carefully distinguish the roles of δ and ε. This is not only essential to achieve tight bounds,but also sensible in practical situations, since it allows one to adapt the bound according to sampledensity and the amount of noise present in the sample separately.
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Dates et versions

hal-04628805 , version 1 (01-07-2024)

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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. SoCG 2024 - 40th International Symposium on Computational Geometry, Jun 2024, Athènes, Greece. ⟨10.4230/LIPIcs.SoCG.2024.11⟩. ⟨hal-04628805⟩
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