  The manifold Helmholtzian (1-Laplacian) operator $\Delta_1$ 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 $\Delta_1$.

Machine Learning;Machine Learning