Hypoelliptic Diffusion Maps I: Tangent Bundles

TL;DR

This paper introduces Hypoelliptic Diffusion Maps (HDM), a novel framework extending manifold learning techniques to analyze data with attached local structures, revealing connections with sub-Riemannian geometry and hypoelliptic operators.

Contribution

It generalizes Diffusion Maps to incorporate local geometric structures at each node, specifically analyzing tangent bundles and linking to hypoelliptic differential operators.

Findings

HDM effectively captures local geometric information.

Connections established between HDM and hypoelliptic operators.

Framework applicable to tangent bundles and potentially other fiber bundles.

Abstract

We introduce the concept of Hypoelliptic Diffusion Maps (HDM), a framework generalizing Diffusion Maps in the context of manifold learning and dimensionality reduction. Standard non-linear dimensionality reduction methods (e.g., LLE, ISOMAP, Laplacian Eigenmaps, Diffusion Maps) focus on mining massive data sets using weighted affinity graphs; Orientable Diffusion Maps and Vector Diffusion Maps enrich these graphs by attaching to each node also some local geometry. HDM likewise considers a scenario where each node possesses additional structure, which is now itself of interest to investigate. Virtually, HDM augments the original data set with attached structures, and provides tools for studying and organizing the augmented ensemble. The goal is to obtain information on individual structures attached to the nodes and on the relationship between structures attached to nearby nodes, so as to study the underlying manifold from which the nodes are sampled. In this paper, we analyze HDM on tangent bundles, revealing its intimate connection with sub-Riemannian geometry and a family of hypoelliptic differential operators. In a later paper, we shall consider more general fibre bundles.