几何模糊--Fréchet平均的一种新现象
Geometrical Smeariness -- A new Phenomenon of Fréchet Means
摘要：
在过去的几十年中，中心极限定理（CLT）被推广到非欧数据空间。几年前，人们发现，对于圆上的某些随机变量，如果样本Fréchet均值在其对径点的概率密度为$\frac{1}{2π}$，则样本Fréchet均值在具有非正态分布的指数为$τ< 1/2$的尺度$n^{-τ}$上渐近地围绕总体均值波动。作者和他的合作者最近发现，$τ= 1/6$的一些随机变量在高维领域。在这篇文章中，我们表明，更令人惊讶的是，在更高的维度领域的现象是定性的不同，从圆，因为它完全取决于空间的几何性质，即其曲率，而不是在对径点的密度。这就产生了几何模糊的新概念。因此，球面可以变形，比如说，通过去除平均值的对径点的邻域，并在那里粘上一个平坦空间，并带有一个光滑的过渡段。这在流形上产生了模糊性，流形是同构于欧几里得空间的。我们给出了一个例子家庭的随机变量与2-smeary平均，即与$τ= 1/6$，其范围有一个洞包含切割轨迹的平均值。孔的大小表现出维数灾难，因为它可以随着尺寸的增加，收敛到整个半球相反的地方Fréchet平均。我们观察到模糊的模拟地标形状的肯德尔预形空间和地磁北极位置的二维球体上的真实数据。
英文摘要：
In the past decades, the central limit theorem (CLT) has been generalized to non-Euclidean data spaces. Some years ago, it was found that for some random variables on the circle, the sample Fréchet mean fluctuates around the population mean asymptotically at a scale $n^{-τ}$ with exponent $τ< 1/2$ with a non-normal distribution if the probability density at the antipodal point of the mean is $\frac{1}{2π}$. The author and his collaborator recently discovered that $τ= 1/6$ for some random variables on higher dimensional spheres. In this article we show that, even more surprisingly, the phenomenon on spheres of higher dimension is qualitatively different from that on the circle, as it depends purely on geometrical properties of the space, namely its curvature, and not on the density at the antipodal point. This gives rise to the new concept of geometrical smeariness. In consequence, the sphere can be deformed, say, by removing a neighborhood of the antipodal point of the mean and gluing a flat space there, with a smooth transition piece. This yields smeariness on a manifold, which is diffeomorphic to Euclidean space. We give an example family of random variables with 2-smeary mean, i.e. with $τ= 1/6$, whose range has a hole containing the cut locus of the mean. The hole size exhibits a curse of dimensionality as it can increase with dimension, converging to the whole hemisphere opposite a local Fréchet mean. We observe smeariness in simulated landmark shapes on Kendall pre-shape space and in real data of geomagnetic north pole positions on the two-dimensional sphere.
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