4 research outputs found
Helmholtzian Eigenmap: Topological feature discovery & edge flow learning from point cloud data
The manifold Helmholtzian (1-Laplacian) operator elegantly
generalizes the Laplace-Beltrami operator to vector fields on a manifold
. In this work, we propose the estimation of the manifold
Helmholtzian from point cloud data by a weighted 1-Laplacian . 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
, the Helmholtzian is a useful tool for the analysis of flows and
vector fields on via the Helmholtz-Hodge theorem. In addition, the
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 to