Higher-order Persistence Diagrams
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918493877698560 |
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| author | Fanning, Charles Aktas, Mehmet |
| author_facet | Fanning, Charles Aktas, Mehmet |
| contents | Many topological data analysis (TDA) pipelines compute large collections of persistence diagrams, yet vectorizations and kernel methods discard the rank-induced implication relations among persistence intervals that are essential for faithful structural comparison and interpretability. We introduce higher-order persistence diagrams, a recursive construction in which containment relations among persistence intervals define higher-order persistence intervals. This construction performs comparison and aggregation directly on persistence diagrams and preserves interval-level structure. We use harmonic analysis to reduce frequency-space evaluations of aggregated diagrams to zeta transforms. This reduction avoids explicit construction of higher-order diagrams and replaces quadratic pair enumeration with nearly linear-time evaluation. Experiments on random network models show substantial speedups over explicit aggregation. Anonymized code is available at https://anonymous.4open.science/r/higher-order-persistence-8201. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_09866 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Higher-order Persistence Diagrams Fanning, Charles Aktas, Mehmet Computational Geometry Algebraic Topology Functional Analysis 55N31 F.2.1 Many topological data analysis (TDA) pipelines compute large collections of persistence diagrams, yet vectorizations and kernel methods discard the rank-induced implication relations among persistence intervals that are essential for faithful structural comparison and interpretability. We introduce higher-order persistence diagrams, a recursive construction in which containment relations among persistence intervals define higher-order persistence intervals. This construction performs comparison and aggregation directly on persistence diagrams and preserves interval-level structure. We use harmonic analysis to reduce frequency-space evaluations of aggregated diagrams to zeta transforms. This reduction avoids explicit construction of higher-order diagrams and replaces quadratic pair enumeration with nearly linear-time evaluation. Experiments on random network models show substantial speedups over explicit aggregation. Anonymized code is available at https://anonymous.4open.science/r/higher-order-persistence-8201. |
| title | Higher-order Persistence Diagrams |
| topic | Computational Geometry Algebraic Topology Functional Analysis 55N31 F.2.1 |
| url | https://arxiv.org/abs/2605.09866 |