Lifting Cocycles: From Heuristic to Theory
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914047408996352 |
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| author | Gaukstad, Sigurd Nordal, Mathias Karsrud Thaule, Marius |
| author_facet | Gaukstad, Sigurd Nordal, Mathias Karsrud Thaule, Marius |
| contents | The circular coordinates algorithm, a key tool in topological data analysis, relies on a theoretically unvalidated lifting step to convert cocycles from a prime field to integer coefficients. We provide a rigorous analysis of this procedure, establishing a criterion for its success. We also introduce a novel algebraic method to reduce any lifted cocycle to a cocycle with winding number 1, ensuring feature correctness. These principles are extended to homology cycles, solidifying the theoretical foundation of this widely used feature extraction technique. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_16102 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lifting Cocycles: From Heuristic to Theory Gaukstad, Sigurd Nordal, Mathias Karsrud Thaule, Marius Algebraic Topology Computational Geometry Primary 55N99, Secondary 68U05 The circular coordinates algorithm, a key tool in topological data analysis, relies on a theoretically unvalidated lifting step to convert cocycles from a prime field to integer coefficients. We provide a rigorous analysis of this procedure, establishing a criterion for its success. We also introduce a novel algebraic method to reduce any lifted cocycle to a cocycle with winding number 1, ensuring feature correctness. These principles are extended to homology cycles, solidifying the theoretical foundation of this widely used feature extraction technique. |
| title | Lifting Cocycles: From Heuristic to Theory |
| topic | Algebraic Topology Computational Geometry Primary 55N99, Secondary 68U05 |
| url | https://arxiv.org/abs/2509.16102 |