Diff. Geom, Math. Phys., PDE Seminar: Micah Warren

Let M be a submanifold of Euclidean space, and let X be a subset of N points, randomly sampled. Belkin and Niyogi showed that one can recover the Laplacian on functions on M as N gets large, by integrating the heat kernel. More recently, Singer and Wu use Principle Component Analysis to construct connection matrices between approximate tangent spaces for nearby points in X. This allows them to construct a rough Laplacian on 1-forms. Together with Ache, we show that by iterating the Laplace operator of Belkin and Niyogi, a la Bakry and Emery, and appealing to the Bochner formula, we can reconstruct the Ricci curvature on the approximate tangent spaces. Combining our work with the work of Singer and Wu, we are able to approximate the Hodge Laplacian on 1-forms.