We give an asymptotic expansion of the relative entropy between the heat kernel of a compact Riemannian manifold and the normalized Riemannian volume for small values of and for a fixed element . 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 , 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.

中文翻译：

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黎曼流形上热核熵的小时间渐近

对于固定元素 和 的小值，我们给出紧致黎曼流形的热核和归一化黎曼体积之间相对熵的渐近展开。我们证明，当用法坐标表示时，展开式中的系数可以表示为曲率张量及其协变导数分量中的通用多项式。我们描述了一种计算系数的方法，并使用该方法来计算前三个系数。渐近展开对于称为扩散变分自动编码器的无监督机器学习算法是必要的。