Manifold Learning with Arbitrary Norms

Manifold learning methods play a prominent role in nonlinear dimensionality reduction and other tasks involving high-dimensional data sets with low intrinsic dimensionality. Many of these are graph-based: they associate vertex each point weighted edge pair. Existing theory shows that the Laplacian matrix graph converges to Laplace–Beltrami operator manifold, under assumption pairwise affinities based on Euclidean norm. In this paper, we determine limiting differential for Laplacians constructed using any Our proof involves an interplay between second fundamental form manifold convex geometry given norm’s unit ball. To demonstrate potential benefits non-Euclidean norms learning, consider task mapping motion large molecules continuous variability. numerical simulation show modified eigenmaps algorithm, Earthmover’s distance, outperforms classic eigenmaps, both terms computational cost sample size needed recover geometry.