Susovan Pal (Mathematics and Data Science Unit (WIDS), Vrije Universiteit Brussel): Asymptotics of the graph Laplacian on manifolds with non-smooth boundaries and their applications.

Manifold learning algorithms often assume that data lie on or near a smooth lower-dimensional manifold M embedded in a higher dimensional Euclidean space, and that the Laplace-Beltrami operator of M can be approximated by graph Laplacian constructed from the data. Analogous results for singular spaces (for instance, spaces with boundaries or cusps) remain largely unexplored. In this talk, I will present my recent work with David Tewodrose analyzing the asymptotic behavior of the graph Laplacian on manifolds with non-smooth boundaries, which we refer to as manifolds with kinks, corners or cusps being special cases. In contrast with the smooth case-where convergence is to the Laplace-Beltrami operator-we show that the limiting behavior involves a first-order boundary operator, namely a generalized normal derivative. Numerical simulations support and illustrate the theoretical results. Aside from the usual motivation of nonlinear dimensionality reduction, we also show one application of this pointwise convergence on edge detection in image processing.