Heat and Matérn Kernels on Matchings

2026-04-15

AI Summary

Constructing geometrically aware kernel functions on discrete non-Euclidean structures such as matchings poses significant challenges. This work presents the first systematic theory of stationary kernels on the space of matchings, extending widely used Euclidean kernels—specifically the heat and Matérn families—to this setting. The authors develop a sub-exponential time algorithm based on zonal polynomials, substantially overcoming the previously prohibitive super-exponential computational complexity. Furthermore, they generalize their framework to the space of phylogenetic trees, elucidating theoretical limitations of such kernel methods and posing new open questions. This study lays foundational groundwork for scalable kernel-based learning on discrete structured domains.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 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.

Problem

• matchings
• kernels
• non-Euclidean geometry
• computational complexity
• phylogenetic trees

Innovation

• geometric kernels
• matchings
• heat kernel
• Matérn kernel
• zonal polynomials