匹配空间上的热核与Matérn核
Heat and Matérn Kernels on Matchings
AI总结：
本文提出了一种构建尊重匹配空间自然几何的几何核的方法，通过热核和Matérn核家族实现非欧几里得空间的核方法，并引入高效算法解决计算成本问题，同时探索其在树空间中的应用。
摘要：
将核方法应用于匹配空间具有挑战性，因为其离散且非欧几里得的性质。本文发展了一个原理性的框架，用于构建尊重匹配空间自然几何的几何核。首先，我们提供了一个完整的平稳核特征化，即尊重该空间固有对称性的核。由于平稳核类过于广泛，我们特别关注热核和Matérn核家族，加入适当的诱导偏差以实现平滑性。虽然这些家族成功地将广泛使用的欧几里得核扩展到匹配空间，但直接评估它们会带来指数级的计算成本。为克服这一困难，我们引入并分析了一种新的亚指数算法，利用区化多项式实现高效的核评估。最后，受匹配与系统发育树之间已知的双射对应关系的启发，我们探索该框架是否能无缝转移到树空间，建立新的负结果并识别一个重要的开放问题。
英文摘要：
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.
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