Manifolds with kinks and the asymptotic behavior of the graph Laplacian operator with Gaussian kernel

TL;DR

This paper studies the asymptotic behavior of the graph Laplacian with Gaussian kernel on manifolds with kinks, including singular boundaries, revealing how local geometry influences the operator's limit.

Contribution

It introduces manifolds with kinks and derives the asymptotic behavior of the graph Laplacian on these spaces, extending understanding to singular boundary cases.

Findings

Asymptotic behavior depends on the inward sector of the tangent space.

Derived behavior near interior and singular points.

Numerical simulations validate theoretical results.

Abstract

We introduce manifolds with kinks, a class of manifolds with possibly singular boundary that notably contains manifolds with smooth boundary and corners. We derive the asymptotic behavior of the Graph Laplace operator with Gaussian kernel and its deterministic limit on these spaces as bandwidth goes to zero. We show that this asymptotic behavior is determined by the inward sector of the tangent space and, as special cases, we derive its behavior near interior and singular points. Lastly, we show the validity of our theoretical results using numerical simulation.