Heat and Mat\'ern Kernels on Matchings

TL;DR

This paper introduces a geometric kernel framework for matchings, focusing on heat and Matérn kernels, with efficient evaluation algorithms and applications to phylogenetic trees.

Contribution

It characterizes stationary kernels on matchings, extends heat and Matérn kernels with smoothness bias, and develops a sub-exponential algorithm for their evaluation.

Findings

Heat and Matérn kernels extend Euclidean kernels to matchings.

Efficient sub-exponential algorithm for kernel evaluation using zonal polynomials.

Exploration of kernel transferability to phylogenetic trees with negative results.

Abstract

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 Mat\'{e}rn 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.