Small time asymptotics of the entropy of the heat kernel on a Riemannian manifold

TL;DR

This paper derives an asymptotic expansion for the relative entropy of the heat kernel on a compact Riemannian manifold, expressing coefficients via curvature tensors, aiding machine learning algorithms.

Contribution

It provides a method to compute the coefficients of the heat kernel entropy expansion in terms of curvature tensors and their derivatives.

Findings

First three coefficients explicitly computed.

Coefficients expressed as universal polynomials in curvature components.

Expansion applicable to machine learning algorithms like Diffusion Variational Autoencoder.

Abstract

We give an asymptotic expansion of the relative entropy between the heat kernel $q_Z(t,z,w)$ of a compact Riemannian manifold $Z$ and the normalized Riemannian volume for small values of $t$ and for a fixed element $z\in Z$. We prove that coefficients in the expansion can be expressed as universal polynomials in the components of the curvature tensor and its covariant derivatives at $z$, when they are expressed in terms of normal coordinates. We describe a method to compute the coefficients, and we use the method to compute the first three coefficients. The asymptotic expansion is necessary for an unsupervised machine-learning algorithm called the Diffusion Variational Autoencoder.