Non-Parametric Manifold Learning
This work addresses a theoretical challenge in non-commutative geometry and manifold learning, but it appears incremental as it builds on existing formulas and methods.
The authors tackled the problem of estimating distances on Riemannian manifolds using graph Laplacian estimates, resulting in an error-bound proof for consistency of manifold distances.
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.