Non-Parametric Manifold Learning

We introduce an estimator for manifold distances based on graph Laplacian estimates of the Laplace-Beltrami operator. We show that the estimator is consistent for suitable choices of graph Laplacians in the literature, based on an equidistributed sample of points drawn from a smooth density bounded away from zero on an unknown compact Riemannian submanifold of Euclidean space. The estimator resembles, and in fact its convergence properties are derived from, a special case of the Kontorovic dual reformulation of Wasserstein distance known as Connes' Distance Formula.