流形上的扩散平均与热核
Diffusion Means and Heat Kernel on Manifolds
摘要：
我们引入扩散手段作为流形数据空间上的位置统计。扩散平均值被定义为具有给定扩散率的各向同性扩散的起点。因此，它们可以被定义在布朗运动可以被定义的所有空间上，并且使用热核展开可以在各种空间上对样品扩散平均值进行数值计算。我们提出了几类空间，其中的热核是已知的，因此可以计算样本扩散手段。作为一个例子，我们调查了一个经典的数据集，从方向统计，样本Fréchet平均表现出有限的样本模糊。
英文摘要：
We introduce diffusion means as location statistics on manifold data spaces. A diffusion mean is defined as the starting point of an isotropic diffusion with a given diffusivity. They can therefore be defined on all spaces on which a Brownian motion can be defined and numerical calculation of sample diffusion means is possible on a variety of spaces using the heat kernel expansion. We present several classes of spaces, for which the heat kernel is known and sample diffusion means can therefore be calculated. As an example, we investigate a classic data set from directional statistics, for which the sample Fréchet mean exhibits finite sample smeariness.
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