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

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.