A simple way of obtaining robust estimates of the "center" (or the "location") and of the "spread" of a dataset is to use the maximum likelihood estimate with a class of heavy-tailed distributions, regardless of the "true" distribution generating the data. We observe that the maximum likelihood problem for the Cauchy distributions, which have particularly heavy tails, is geodesically convex and therefore efficiently solvable (Cauchy distributions are parametrized by the upper half plane, i.e. by the hyperbolic plane). Moreover, it has an appealing geometrical meaning: the datapoints, living on the boundary of the hyperbolic plane, are attracting the parameter by unit forces, and we search the point where these forces are in equilibrium. This picture generalizes to several classes of multivariate distributions with heavy tails, including, in particular, the multivariate Cauchy distributions. The hyperbolic plane gets replaced by symmetric spaces of noncompact type. Geodesic convexity gives us an efficient numerical solution of the maximum likelihood problem for these distribution classes. This can then be used for robust estimates of location and spread, thanks to the heavy tails of these distributions.

翻译：获取数据集的“中心”（或“位置”）和“离散度”的稳健估计的一种简单方法是使用一类重尾分布的最大似然估计，而无论生成数据的“真实”分布是什么。我们观察到，具有特别重尾的Cauchy分布的最大似然问题是测地凸的，因此可以高效求解（Cauchy分布由上半平面即双曲平面参数化）。此外，这一问题具有引人注目的几何含义：位于双曲平面边界上的数据点通过单位力吸引参数，我们寻找这些力达到平衡的点。这一图景可推广至多类重尾多元分布，特别是多元Cauchy分布。双曲平面被非紧型对称空间替代。测地凸性使我们能够对这些分布类的最大似然问题给出高效的数值解。由于这些分布的重尾特性，该结果可用于位置和离散度的稳健估计。