Manifold learning with arbitrary norms

TL;DR

This paper extends manifold learning theory to arbitrary norms, showing the limiting differential operator for graph Laplacians and demonstrating benefits of non-Euclidean norms in molecular motion mapping.

Contribution

It generalizes the convergence theory of graph Laplacians to any norm and introduces a modified Laplacian eigenmaps algorithm using Earthmover's distance.

Findings

The limiting differential operator is characterized for any norm.

Non-Euclidean norms can improve manifold learning performance.

Earthmover's distance-based Laplacian eigenmaps outperform Euclidean methods.

Abstract

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