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 Fr\'echet mean exhibits finite sample smeariness.
翻译:我们引入了传播手段,作为多个数据空间的定位统计。扩散手段被定义为通过给定的分辨度进行等热带扩散的起点。因此,可以在所有空间进行定义,在这些空间上可以界定布朗运动,可以使用热内核扩展的各种空间对样本扩散手段进行数字计算。我们呈现出若干类空间,这些空间的热内核已知,因此可以计算出样本扩散手段。举例来说,我们从方向统计中调查了一套经典数据,对此,Fr\'echet的样本意味着有一定的样本模糊性。