Helmholtzian Eigenmap: Topological feature discovery & edge flow learning from point cloud data

The manifold Helmholtzian (1-Laplacian) operator Delta1 elegantly generalizes the Laplace-Beltrami operator to vector fields on a manifold mathcal M. In this work, we propose the estimation of the manifold Helmholtzian from point cloud data by a weighted 1-Laplacian mathcal L1. While higher order Laplacians have been introduced and studied, this work is the first to present a graph Helmholtzian constructed from a simplicial complex as a consistent estimator for the continuous operator in a non-parametric setting. Equipped with the geometric and topological information about mathcal M, the Helmholtzian is a useful tool for the analysis of flows and vector fields on mathcal M via the Helmholtz-Hodge theorem. In addition, the mathcal L1 allows the smoothing, prediction, and feature extraction of the flows. We demonstrate these possibilities on substantial sets of synthetic and real point cloud datasets with non-trivial topological structures; and provide theoretical results on the limit of mathcal L1 to Delta1.