Non-Parametric Manifold Learning

We introduce an estimator for distances in a compact Riemannian manifold M based on graph Laplacian estimates of the Laplace-Beltrami operator. We upper bound the l2-loss for the ratio of the estimator over the true manifold distance, or more precisely an approximation of manifold distance in non-commutative geometry (cf. [Connes and Suijelekom, 2020]), in terms of spectral errors in the graph Laplacian estimates and, implicitly, several geometric properties of the manifold. We consequently obtain a consistency result for the estimator for samples equidistributed from a strictly positive density on M and graph Laplacians which spectrally converge, in a suitable sense, to the Laplace-Beltrami operator. 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.