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The manifold Helmholtzian (1-Laplacian) operator $\Delta_1$ elegantlygeneralizes the Laplace-Beltrami operator to vector fields on a manifold$\mathcal M$. In this work, we propose the estimation of the manifoldHelmholtzian from point cloud data by a weighted 1-Laplacian $\mathcal L_1$.While higher order Laplacians have been introduced and studied, this work isthe first to present a graph Helmholtzian constructed from a simplicial complexas a consistent estimator for the continuous operator in a non-parametricsetting. Equipped with the geometric and topological information about$\mathcal M$, the Helmholtzian is a useful tool for the analysis of flows andvector fields on $\mathcal M$ via the Helmholtz-Hodge theorem. In addition, the$\mathcal L_1$ allows the smoothing, prediction, and feature extraction of theflows. We demonstrate these possibilities on substantial sets of synthetic andreal point cloud datasets with non-trivial topological structures; and providetheoretical results on the limit of $\mathcal L_1$ to $\Delta_1$.

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