Non-Parametric Manifold Learning

TL;DR

This paper proposes a new non-parametric estimator for manifold distances using graph Laplacian estimates, providing bounds on errors and proving consistency, with implications for non-commutative geometry and Wasserstein distances.

Contribution

It introduces a novel estimator for manifold distances based on graph Laplacian estimates, with theoretical error bounds and consistency proofs.

Findings

Bounded the error in manifold distance estimates in terms of spectral errors.

Proved the estimator's consistency for untruncated manifold distances.

Connected the estimator's properties to Connes' Distance Formula and Wasserstein distances.

Abstract

We introduce an estimator for distances in a compact Riemannian manifold based on graph Laplacian estimates of the Laplace-Beltrami operator. We upper bound the error in the estimate of manifold distances, or more precisely an estimate of a spectrally truncated variant of manifold distance of interest 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. A consequence is a proof of consistency for (untruncated) manifold distances. 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.