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

## TL;DR

This paper derives an exact heat kernel on a hypersphere and demonstrates its improved effectiveness over heuristic kernels in kernel SVMs across various applications.

## Contribution

It provides a higher order parametrix expansion for the hyperspherical heat kernel and compares it with heuristic versions, showing enhanced performance.

## Key findings

- Exact kernel often outperforms heuristic in SVM classification
- Higher order expansion improves kernel accuracy
- Applications include text mining, mutation imputation, stock analysis

## Abstract

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.