论文标题
在riemannian歧管上热核的熵的少量时间渐近。
Small time asymptotics of the entropy of the heat kernel on a Riemannian manifold
论文作者
论文摘要
我们给出了紧凑的Riemannian歧管$ z $的热核$ q_z（t，z，w）之间的相对熵的渐近扩展，而对于固定元素$ z \ in z $中的固定元素$ z \，均值的riemannian量为formanized riemannian量。我们证明，扩展中的系数可以表示为曲率张量的组件中的通用多项式及其在$ z $的协方差衍生物，当它们用正常坐标表示时。我们描述了一种计算系数的方法，并使用该方法计算前三个系数。对于一种无监督的机器学习算法，渐近扩展是必需的，称为扩散变化自动编码器。
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.
