# Exact heat kernel on a hypersphere and its applications in kernel SVM

arXiv: https://arxiv.org/abs/1702.01373 · HTML: https://mlsanity.com/paper/1702.01373/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Feb 2017.

## Built on by (most cited first, 7 of 7)

- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case](https://mlsanity.com/paper/2208.14960/) (2022)
- [Expanding SPHERE-JEPA: A Family of Statistical Regularizers for the Hypersphere](https://mlsanity.com/paper/2606.17603/) (2026)
- [Manifold Random Features](https://mlsanity.com/paper/2602.03797/) (2026)
- [Beyond Gaussian Worlds: Latent Geometry Matters for JEPAs](https://mlsanity.com/paper/2609.21656/) (2026)
- [Probability-Generating Function Kernels for Spherical Data](https://mlsanity.com/paper/2112.00365/) (2021)
- [Conformal Transformation of Kernels: A Geometric Perspective on Text Classification](https://mlsanity.com/paper/2406.00499/) (2024)
- [Manifold Diffusion for Structure Generation of Transition Metal Complexes](https://mlsanity.com/paper/2606.00666/) (2026)

## Abstract (arXiv)

Many contemporary statistical learning methods assume a Euclidean feature space. This paper presents a method for defining similarity based on hyperspherical geometry and shows that it often improves the performance of support vector machine compared to other competing similarity measures. Specifically, the idea of using heat diffusion on a hypersphere to measure similarity has been previously proposed, demonstrating promising results based on a heuristic heat kernel obtained from the zeroth order parametrix expansion; however, how well this heuristic kernel agrees with the exact hyperspherical heat kernel remains unknown. This paper presents a higher order parametrix expansion of the heat kernel on a unit hypersphere and discusses several problems associated with this expansion method. We then compare the heuristic kernel with an exact form of the heat kernel expressed in terms of a uniformly and absolutely convergent series in high-dimensional angular momentum eigenmodes. Being a natural measure of similarity between sample points dwelling on a hypersphere, the exact kernel often shows superior performance in kernel SVM classifications applied to text mining, tumor somatic mutation imputation, and stock market analysis.