Helmholtzian Eigenmap: Topological feature discovery & edge flow learning from point cloud data
This provides a tool for topological feature discovery and edge flow learning in computational geometry and data analysis, but it is incremental as it builds on existing Laplacian methods.
They tackled the problem of estimating the manifold Helmholtzian operator from point cloud data to analyze flows and vector fields, achieving a consistent estimator with theoretical convergence results and demonstrating applications on synthetic and real datasets.
The manifold Helmholtzian (1-Laplacian) operator $Δ_1$ 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 $\mathcal L_1$. 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 $\mathcal L_1$ 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 $\mathcal L_1$ to $Δ_1$.