Local Data-Driven Exterior Calculus and Hodge Laplacian Approximations on Point Clouds

We introduce a local, data-driven framework for approximating differential forms, exterior derivatives, and Hodge Laplacians directly from unstructured point cloud data sampled from an unknown compact Riemannian manifold. The proposed methodology requires neither meshes, simplicial complexes, nor global spectral decompositions; instead, it relies on local tensor-array representations and convolution-type operators constructed from neighboring sample points.We first develop computational representations of differential $k$-forms through alternating tensor arrays naturally associated with local tangent-space coordinates. Using these representations, we construct a local approximation of the exterior derivative via kernel interactions among nearby sample points, and prove both pointwise and uniform convergence of the proposed operator toward the smooth exterior derivative as the scale parameter tends to zero.Building on this construction, we derive explicit matrix representations of the Hodge $k$-Laplacians through an alternating array formalism and prove weak convergence toward the continuous Hodge Laplacian. As applications, we develop a data-driven numerical scheme for the heat equation on differential forms via a forward Euler discretization of the Hodge Laplacian matrix, and introduce the \emph{Hodge distance}, a novel point-to-point metric that encodes the intrinsic differential-geometric structure of the underlying manifold beyond standard Euclidean or geodesic proximity.Numerical experiments on the two-dimensional sphere~$\mathbb{S}^2$ validate the theoretical predictions and illustrate the geometric sensitivity of the Hodge distance to the intrinsic topology of the manifold.