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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.03163 |
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| _version_ | 1866917244441722880 |
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| author | Lentz, Christian Henselman-Petrusek, Gregory Ziegelmeier, Lori |
| author_facet | Lentz, Christian Henselman-Petrusek, Gregory Ziegelmeier, Lori |
| contents | A central problem in data-driven scientific inquiry is how to interpret structure in noisy, high-dimensional data. Topological data analysis (TDA) provides a solution via persistent homology, which encodes features of interest as topological holes within a filtration of data. The present work extends this framework to a related invariant which uncovers topological structure of a space relative to a subspace: persistent relative homology (PRH). We show that this invariant can be computed in a simple, highly transparent and general manner, using a two-step matrix reduction technique with worst-case time complexity comparable to ordinary persistent homology. We provide proofs demonstrating the correctness and computational complexity of this approach in addition to a performance-optimized implementation for a special case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03163 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A U-match Algorithm for Persistent Relative Homology Lentz, Christian Henselman-Petrusek, Gregory Ziegelmeier, Lori Algebraic Topology A central problem in data-driven scientific inquiry is how to interpret structure in noisy, high-dimensional data. Topological data analysis (TDA) provides a solution via persistent homology, which encodes features of interest as topological holes within a filtration of data. The present work extends this framework to a related invariant which uncovers topological structure of a space relative to a subspace: persistent relative homology (PRH). We show that this invariant can be computed in a simple, highly transparent and general manner, using a two-step matrix reduction technique with worst-case time complexity comparable to ordinary persistent homology. We provide proofs demonstrating the correctness and computational complexity of this approach in addition to a performance-optimized implementation for a special case. |
| title | A U-match Algorithm for Persistent Relative Homology |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2602.03163 |