Der-Chen Chang, Ophir Frieder, Chi-Feng Hung, Hao-Ren Yao, Heat kernels on unit spheres and applications to graph kernels

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DOI: 10.23952/jnva.7.2023.3.01

Volume 7, Issue 3, 1 June 2023, Pages 309-343

Abstract. It is known that many statistical and machine learning approaches heavily rely on pairwise distance between data points. The choice of distance function on the underlying manifold has a fundamental impact on performance of these processes. This is closely related to questions of how to appropriately calculate distances, and hence, fundamental solutions (heat kernels) for heat operators can be obtained. In general, it is not so easy to obtain a closed form for heat kernels. We first survey results of heat kernels on radially symmetric Riemannian manifolds, e.g., Euclidean spaces and unit spheres in ${\mathbf R}^n$ . For the cases $n=1,2,3$ , we may construct the heat kernel explicitly. But, the computation is much more complicated when $n>3$ . However, by results of Nagase, we may construct parametrices for the heat kernel by using elementary functions so that the error terms can be under controlled. In the second part of the paper, we discuss some results on subRiemannian manifolds, especially $3$ -dimensional sphere in ${\mathbf C}^2$ as a CR-manifold. We study geodesics connecting two given points on $\mathbb S^3$ respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of $\mathbb S^3$ and some partial results are obtained in the general case. The Carnot-Carathéodory distance is calculated. We also present some motivations related to quantum mechanics. Then we give a brief discussion of Greiner’s methods on the heat kernel for the Cauchy-Riemann subLaplacian on $\mathbb S^{2n+1}$ . We provide a brief discussion on applications of these heat kernels to graph kernels in the last part of the paper.

How to Cite this Article:
D.C. Chang, O. Frieder, C.F. Hung, H.R. Yao, Heat kernels on unit spheres and applications to graph kernels, J. Nonlinear Var. Anal. 7 (2023), 309-343.