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.

中文翻译：

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光谱外微积分

开发了一种在流形学习问题中构建外部微积分的谱方法。谱方法被证明在大数据的限制下收敛到真正的外部微积分。同时，谱方法将内存需求与数据点数量和环境空间维度分离。为了实现这一点，外部微积分完全根据函数的拉普拉斯算子的特征值和特征函数重新表述。这些特征函数的外部导数（和它们的楔积）被证明为合适的 $k$-形式的 $L^2$ 空间以及高阶 Sobolev 空间形成一个框架（一种生成集）。推导出公式以根据函数上的拉普拉斯算子的特征函数和特征值来表达形式上的拉普拉斯-德拉姆算子。通过在该框架中表示 Laplace-de Rham 算子，可以通过 Galerkin 近似技术获得谱收敛结果。数值例子证明了 Laplace-de Rham 算子在 1-形式上的特征值和特征形式的准确恢复。正确的 Betti 数是从这个算子的内核中获得的，该内核是从几个可定向和不可定向流形上采样的数据中近似得出的，并且特征形式通过它们相应的向量场进行可视化。这些向量场形成平方可积向量场空间的自然正交基，并由测量振荡行为的狄利克雷能量泛函排序。光谱框架还在非光滑示例（Lorenz 63 吸引子）上显示出有希望的结果，