Helmholtzian Eigenmap: Topological feature discovery edge flow learning from point cloud data
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The manifold Helmholtzian (1-Laplacian) operator Δ_1 elegantly generalizes the Laplace-Beltrami operator to vector fields on a manifold ℳ. In this work, we propose the estimation of the manifold Helmholtzian from point cloud data by a weighted 1-Laplacian ℒ_1. While higher order Laplacians ave been introduced and studied, this work is the first to present a graph Helmholtzian constructed from a simplicial complex as an estimator for the continuous operator in a non-parametric setting. Equipped with the geometric and topological information about ℳ, the Helmholtzian is a useful tool for the analysis of flows and vector fields on ℳ via the Helmholtz-Hodge theorem. In addition, the 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 ℒ_1 to Δ_1.
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