  A spectral approach to building the exterior calculus in manifold learning
problems is developed. The spectral approach is shown to converge to the true
exterior calculus in the limit of large data. Simultaneously, the spectral
approach decouples the memory requirements from the amount of data points and
ambient space dimension. To achieve this, the exterior calculus is reformulated
entirely in terms of the eigenvalues and eigenfunctions of the Laplacian
operator on functions. The exterior derivatives of these eigenfunctions (and
their wedge products) are shown to form a frame (a type of spanning set) for
appropriate $L^2$ spaces of $k$-forms, as well as higher-order Sobolev spaces.
Formulas are derived to express the Laplace-de Rham operators on forms in terms
of the eigenfunctions and eigenvalues of the Laplacian on functions. By
representing the Laplace-de Rham operators in this frame, spectral convergence
results are obtained via Galerkin approximation techniques. Numerical examples
demonstrate accurate recovery of eigenvalues and eigenforms of the Laplace-de
Rham operator on 1-forms. The correct Betti numbers are obtained from the
kernel of this operator approximated from data sampled on several orientable
and non-orientable manifolds, and the eigenforms are visualized via their
corresponding vector fields. These vector fields form a natural orthonormal
basis for the space of square-integrable vector fields, and are ordered by a
Dirichlet energy functional which measures oscillatory behavior. The spectral
framework also shows promising results on a non-smooth example (the Lorenz 63
attractor), suggesting that a spectral formulation of exterior calculus may be
feasible in spaces with no differentiable structure.

Differential Geometry