高效提取高阶拓扑特征，让复杂数据分析更实用。
Top-K Exterior Power Persistent Homology: Algorithm, Structure, and Stability
- 提出基于锚点流的结构分解法，支持快速找最长区间。
- Top-K结果对输入扰动稳定，理论证明其2-Lipschitz性。
- 适合处理大规模数据的机器学习与科学计算场景。
外幂在计算几何中的持久同调中起关键作用。本文研究从一个拟紧持久模的外幂层中提取前K个最长区间的問題。我们证明了一个结构分解定理，将外幂层组织为具有明确重数的单调锚点流，从而支持最佳优先算法。同时，我们证明了Top-K长度向量在输入条形图的瓶颈扰动下具有2-利普希茨性质，并建立了比较模型下的下界。实验验证了理论正确性，在高重叠情况下相比全枚举有显著加速。该方法使高阶持久同调适用于大规模数据，广泛应用于机器学习、数据科学和科学计算。
原文摘要 · Abstract (English)
Exterior powers play important roles in persistent homology in computational geometry. In the present paper we study the problem of extracting the $K$ longest intervals of the exterior-power layers of a tame persistence module. We prove a structural decomposition theorem that organizes the exterior-power layers into monotone per-anchor streams with explicit multiplicities, enabling a best-first algorithm. We also show that the Top-$K$ length vector is $2$-Lipschitz under bottleneck perturbations of the input barcode, and prove a comparison-model lower bound. Our experiments confirm the theory, showing speedups over full enumeration in high overlap cases. By enabling efficient extraction of the most prominent features, our approach makes higher-order persistence feasible for large datasets and thus broadly applicable to machine learning, data science, and scientific computing.
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