采样 Vietoris-Rips 复形上的上同调类几何
The Geometry of Cochains on Sampled Vietoris-Rips Complexes
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中文总结 AI 辅助
该研究针对紧嵌入流形的独立同分布样本构建的 Vietoris-Rips 复形，定义上同调映射与核加权内积，证明相关量的收敛性并得到离散 Hodge 拉普拉斯算子谱上界等结果。
中文摘要 AI 辅助
我们研究由紧嵌入流形的独立同分布样本构建的 Vietoris-Rips 复形上的单纯上同调类几何。通过在仿射单形上积分微分形式，我们定义了从光滑形式到单纯上同调类的上同调映射，并根据环境两两距离为后者赋予核加权内积。在采样复形恢复流形同伦型的尺度下，我们以高概率证明这些内积的定量收敛性，且在均匀采样下，相关余微分能量收敛到其连续对应量。作为推论，我们得到离散 Hodge 拉普拉斯算子的谱上界；对于可定向流形，还得到离散化谐一形式的谐代表元收敛性以及圆坐标的谐平滑一致性。
英文摘要
We study the geometry of simplicial cochains on Vietoris-Rips complexes built from i.i.d. samples of a compact embedded manifold. By integrating differential forms over affine simplices, we define a cochain map from smooth forms to simplicial cochains and equip the latter with kernel-weighted inner products determined by ambient pairwise distances. At scales where the sampled complex recovers the homotopy type of the manifold, we prove quantitative high-probability convergence of these inner products and, under uniform sampling, of the associated codifferential energies to their continuum counterparts. As consequences, we obtain spectral upper bounds for the discrete Hodge Laplacian and, for orientable manifolds, convergence of harmonic representatives of discretized harmonic one-forms and consistency of harmonic smoothing for circular coordinates.
