Heat and Matérn Kernels on Matchings
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917411774529536 |
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| author | Eremeev, Dmitry Said, Salem Borovitskiy, Viacheslav |
| author_facet | Eremeev, Dmitry Said, Salem Borovitskiy, Viacheslav |
| contents | Applying kernel methods to matchings is challenging due to their discrete, non-Euclidean nature. In this paper, we develop a principled framework for constructing geometric kernels that respect the natural geometry of the space of matchings. To this end, we first provide a complete characterization of stationary kernels, i.e. kernels that respect the inherent symmetries of this space. Because the class of stationary kernels is too broad, we specifically focus on the heat and Matérn kernel families, adding an appropriate inductive bias of smoothness to stationarity. While these families successfully extend widely popular Euclidean kernels to matchings, evaluating them naively incurs a prohibitive super-exponential computational cost. To overcome this difficulty, we introduce and analyze a novel, sub-exponential algorithm leveraging zonal polynomials for efficient kernel evaluation. Finally, motivated by the known bijective correspondence between matchings and phylogenetic trees-a crucial data modality in biology-we explore whether our framework can be seamlessly transferred to the space of trees, establishing novel negative results and identifying a significant open problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14331 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Heat and Matérn Kernels on Matchings Eremeev, Dmitry Said, Salem Borovitskiy, Viacheslav Machine Learning Applying kernel methods to matchings is challenging due to their discrete, non-Euclidean nature. In this paper, we develop a principled framework for constructing geometric kernels that respect the natural geometry of the space of matchings. To this end, we first provide a complete characterization of stationary kernels, i.e. kernels that respect the inherent symmetries of this space. Because the class of stationary kernels is too broad, we specifically focus on the heat and Matérn kernel families, adding an appropriate inductive bias of smoothness to stationarity. While these families successfully extend widely popular Euclidean kernels to matchings, evaluating them naively incurs a prohibitive super-exponential computational cost. To overcome this difficulty, we introduce and analyze a novel, sub-exponential algorithm leveraging zonal polynomials for efficient kernel evaluation. Finally, motivated by the known bijective correspondence between matchings and phylogenetic trees-a crucial data modality in biology-we explore whether our framework can be seamlessly transferred to the space of trees, establishing novel negative results and identifying a significant open problem. |
| title | Heat and Matérn Kernels on Matchings |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2604.14331 |