AI 中文总结
研究Rips和Dowker - Rips复形，针对其缺乏规范原对偶胞腔结构问题，通过配备几何诱导霍奇星算子解决，核心方法是设计权重，如单纯形体积权重和软见证权重，主要贡献是证明相关性质并给出谱描述符用于比较霍奇谱。
AI 中文摘要
Vietoris - Rips复形$\mathrm{VR}_\epsilon(X)$、Dowker复形$\mathrm{D}_R(X,Y)$及其旗化的Dowker - Rips变体$\mathrm{DR}_R(X,Y)=\mathrm{F}(\mathrm{D}_R(X,Y))$是由度量数据或见证关系构建的单纯复形。它们在拓扑数据分析中有用，但固定尺度下保留的底层几何信息少。不像其他复形，Rips型复形没有规范的原对偶胞腔结构。我们通过为Rips型复形$K$配备由正单纯形权重$W_k=\operatorname{diag}\{w_k(\sigma):\sigma\in K_k\}$表示的对角几何诱导霍奇星算子来解决此问题，其定义了$k$ -上链上的加权内积。所得加权离散霍奇拉普拉斯算子$\Delta_k^W$的核维数等于底层复形的第$k$个贝蒂数，非零谱由所选几何权重控制。核心问题是权重设计，我们关注基于欧几里得单纯形体积的单纯形体积权重和基于Dowker式支持函数的软见证权重。我们证明了任意正对角权重的正性、加权自伴性和贝蒂数保持性，建立了软见证支持的渐近衰减率特征，并描述了从$\Delta_k^W$导出的谱描述符以比较Rips复形上的几何感知霍奇谱。
英文摘要
The Vietoris--Rips complex $\mathrm{VR}_ε(X)$, the Dowker complex $\mathrm{D}_R(X,Y)$, and its flagified Dowker--Rips variant $\mathrm{DR}_R(X,Y)=\mathrm{F}(\mathrm{D}_R(X,Y))$ are simplicial complexes constructed from metric data or witness relations. They are useful in topological data analysis because they encode topology through combinatorial data derived from pairwise information, but at a fixed scale they retain little of the underlying geometry. Unlike alpha complexes or mesh-based discretizations, Rips-type complexes carry no canonical primal--dual cell structure, which is the ingredient used by the discrete exterior calculus Hodge star to encode metric information. We address this gap by equipping a Rips-type complex $K$ with diagonal geometry-induced Hodge stars represented by positive simplex weights $W_k=\operatorname{diag}\{w_k(σ):σ\in K_k\}$, which define weighted inner products on $k$-cochains. The resulting weighted discrete Hodge Laplacian $Δ_k^W$ has kernel dimension equal to the $k$th Betti number of the underlying complex, while its nonzero spectrum is governed by the chosen geometric weights. The central issue is therefore not the existence of a weighted Laplacian, since any positive diagonal weights define one, but the design of weights that encode meaningful metric or witness geometry. We focus on two computable choices: simplex-volume weights, based on Euclidean simplex volumes, and soft witness weights, based on a Dowker-style support function $s_t(σ;Y)$ that quantifies higher-order witness support lost under flagification. We prove positivity, weighted self-adjointness, and Betti-number preservation for arbitrary positive diagonal weights, establish an asymptotic decay-rate characterization for soft witness support, and describe spectral descriptors derived from $Δ_k^W$ for comparing geometry-aware Hodge spectra on Rips complexes.
