Representing two-parameter persistence modules via graphcodes
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909532719939584 |
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| author | Kerber, Michael Russold, Florian |
| author_facet | Kerber, Michael Russold, Florian |
| contents | Graphcodes were recently introduced as a technique to employ two-parameter persistence modules in machine learning tasks (Kerber and Russold, NeurIPS 2024). We show in this work that a compressed version of graphcodes yields a description of a two-parameter module that is equivalent to a presentation of the module. This alternative representation as a graph allows for a simple translation between combinatorics and algebra: connected components of the graphcode correspond to summands of the module and isolated paths correspond to intervals. We demonstrate that graphcodes are useful in practice by speeding-up the task of decomposing a module into indecomposable summands. Also, the graphcode viewpoint allows to devise a simple algorithm to decide whether a persistence module is interval-decomposable in $O(n^4)$ time, which improves on the previous bound of $O(n^{2ω+1})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07368 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representing two-parameter persistence modules via graphcodes Kerber, Michael Russold, Florian Algebraic Topology Representation Theory Graphcodes were recently introduced as a technique to employ two-parameter persistence modules in machine learning tasks (Kerber and Russold, NeurIPS 2024). We show in this work that a compressed version of graphcodes yields a description of a two-parameter module that is equivalent to a presentation of the module. This alternative representation as a graph allows for a simple translation between combinatorics and algebra: connected components of the graphcode correspond to summands of the module and isolated paths correspond to intervals. We demonstrate that graphcodes are useful in practice by speeding-up the task of decomposing a module into indecomposable summands. Also, the graphcode viewpoint allows to devise a simple algorithm to decide whether a persistence module is interval-decomposable in $O(n^4)$ time, which improves on the previous bound of $O(n^{2ω+1})$. |
| title | Representing two-parameter persistence modules via graphcodes |
| topic | Algebraic Topology Representation Theory |
| url | https://arxiv.org/abs/2503.07368 |