Exact Heat Kernel on a Hypersphere and Its Applications in Kernel SVM

Frontiers in Applied Mathematics and Statistics

Song, Jun S.¹  Zhao, Chenchao² 

[1] Carl R. Woese Institute for Genomic Biology, University of Illinois at Urbana-Champaign, United States;Department of Physics, University of Illinois at Urbana-Champaign, United States

关键词: heat kernel;    support vector machine (SVM);    hyperspherical geometry;    Document classification;    Genomics;    time series;   

DOI  :  10.3389/fams.2018.00001

学科分类：数学（综合）

来源: Frontiers

【 摘 要 】

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 and tested by \citet{Lafferty:2015uy}, 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.

【 授权许可】

CC BY