DIffusion Maps for Manifolds with Boundary

Abstract

Diffusion maps and other graph Laplacian based methods in machine learning have a long history of pointwise consistency results. This work provides rigorous formulation for diffusion maps for manifolds with boundary. We show that different normalizations of graph Laplacians are asymptotically unbiased for manifolds with boundary when viewed in a variational sense. A crucial component of this work is the introduction of semigeodesic coordinates, which allow for more systematic treatment of boundary points. In particular, we derive new pointwise expansions which relate first-order error to the mean curvature of the boundary of M.