Manifolds with Non-Smooth Boundaries and Asymptotics of the Graph Laplacian

Susovan Pal, David Tewodrose
Proceedings of the Geometry, Topology, and Machine Learning Workshop, PMLR 325:246-250, 2026.

Abstract

This work studies the asymptotic behavior of discrete graph Laplacians constructed from random samples on Riemannian manifolds whose boundaries may exhibit geometric irregularities. We introduce the class of \emph{manifolds with kinks} (MFK)—a broad generalization of smooth manifolds with boundaries and corners—and establish convergence results of the graph Laplacian at interior, border, and cusp points. The results unify earlier analyses on smooth domains (Belkin–Niyogi, Hein–Luxburg, Peoples–Harlim) and extend them to non-smooth geometries that frequently occur in data analysis. We also discuss applications to edge detection in image processing and possible extensions to curvature-dependent asymptotics.

Cite this Paper

BibTeX
@InProceedings{pmlr-v325-pal26a, title = {Manifolds with Non-Smooth Boundaries and Asymptotics of the Graph Laplacian}, author = {Pal, Susovan and Tewodrose, David}, booktitle = {Proceedings of the Geometry, Topology, and Machine Learning Workshop}, pages = {246--250}, year = {2026}, editor = {Bleher, Michael and Jensen, Freya and Maier, Levin and Taha, Diaaeldin and Wienhard, Anna}, volume = {325}, series = {Proceedings of Machine Learning Research}, month = {10--14 Nov}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v325/main/assets/pal26a/pal26a.pdf}, url = {https://proceedings.mlr.press/v325/pal26a.html}, abstract = {This work studies the asymptotic behavior of discrete graph Laplacians constructed from random samples on Riemannian manifolds whose boundaries may exhibit geometric irregularities. We introduce the class of \emph{manifolds with kinks} (MFK)—a broad generalization of smooth manifolds with boundaries and corners—and establish convergence results of the graph Laplacian at interior, border, and cusp points. The results unify earlier analyses on smooth domains (Belkin–Niyogi, Hein–Luxburg, Peoples–Harlim) and extend them to non-smooth geometries that frequently occur in data analysis. We also discuss applications to edge detection in image processing and possible extensions to curvature-dependent asymptotics.} }
Endnote
%0 Conference Paper %T Manifolds with Non-Smooth Boundaries and Asymptotics of the Graph Laplacian %A Susovan Pal %A David Tewodrose %B Proceedings of the Geometry, Topology, and Machine Learning Workshop %C Proceedings of Machine Learning Research %D 2026 %E Michael Bleher %E Freya Jensen %E Levin Maier %E Diaaeldin Taha %E Anna Wienhard %F pmlr-v325-pal26a %I PMLR %P 246--250 %U https://proceedings.mlr.press/v325/pal26a.html %V 325 %X This work studies the asymptotic behavior of discrete graph Laplacians constructed from random samples on Riemannian manifolds whose boundaries may exhibit geometric irregularities. We introduce the class of \emph{manifolds with kinks} (MFK)—a broad generalization of smooth manifolds with boundaries and corners—and establish convergence results of the graph Laplacian at interior, border, and cusp points. The results unify earlier analyses on smooth domains (Belkin–Niyogi, Hein–Luxburg, Peoples–Harlim) and extend them to non-smooth geometries that frequently occur in data analysis. We also discuss applications to edge detection in image processing and possible extensions to curvature-dependent asymptotics.
APA
Pal, S. & Tewodrose, D.. (2026). Manifolds with Non-Smooth Boundaries and Asymptotics of the Graph Laplacian. Proceedings of the Geometry, Topology, and Machine Learning Workshop, in Proceedings of Machine Learning Research 325:246-250 Available from https://proceedings.mlr.press/v325/pal26a.html.

Related Material