Top-K Exterior Power Persistent Homology: Algorithm, Structure, and Stability
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914217323397120 |
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| author | Maruyama, Yoshihiro |
| author_facet | Maruyama, Yoshihiro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20325 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Top-K Exterior Power Persistent Homology: Algorithm, Structure, and Stability Maruyama, Yoshihiro Computational Geometry Discrete Mathematics Machine Learning 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. |
| title | Top-K Exterior Power Persistent Homology: Algorithm, Structure, and Stability |
| topic | Computational Geometry Discrete Mathematics Machine Learning |
| url | https://arxiv.org/abs/2512.20325 |