Heat and Mat\'ern Kernels on Matchings

2026-04-17 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This paper introduces a principled framework for constructing geometric kernels, specifically Heat and Matérn kernels, for the discrete, non-Euclidean space of matchings. The authors first characterize stationary kernels that respect the inherent symmetries of this space, identifying the space of matchings as a homogeneous space $S_{2n}/H_{n}$. To overcome the prohibitive super-exponential computational cost of naive kernel evaluation, a novel sub-exponential algorithm is developed, leveraging zonal polynomials. This algorithmic advance allows for exact or highly accurate approximate kernel computations for larger problem sizes, such as $n=25$. The framework also explores the connection between matchings and phylogenetic trees, establishing negative results regarding geometry preservation.

Key takeaway

For AI Scientists and Research Scientists working with discrete combinatorial structures like matchings, this framework offers a principled way to apply kernel methods. You should consider adopting the proposed Heat and Matérn kernels, leveraging the sub-exponential algorithm based on zonal polynomials to manage computational complexity, especially for problems with $n$ up to 25. This approach enables robust machine learning applications on non-Euclidean data where traditional methods fail.

Key insights

Geometric kernels for matchings can be constructed by respecting symmetries and incorporating smoothness, despite computational challenges.

Principles

• Stationarity and smoothness are key for generalizable kernels.
• Homogeneous spaces enable characterization of stationary kernels.

Method

A novel sub-exponential algorithm using zonal polynomials evaluates Heat and Matérn kernels on matchings, reducing complexity from super-exponential to sub-exponential in $n$.

In practice

• Apply geometric kernels to discrete combinatorial structures.
• Utilize zonal polynomials for efficient kernel computation.

Topics

• Matchings
• Geometric Kernels
• Heat and Matérn Kernels
• Stationary Kernels
• Zonal Polynomials

Code references

• eremeev-d/matchings-gp

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.