Smeary Limit Theorems for Generalized Fréchet Means on Non-Euclidean Spaces

非欧空间上广义 Fréchet 均值的模糊极限定理

• 批准号: 427894948
• 负责人: Professor Dr. Stephan Huckemann
• 金额: --
• 依托单位: Institut für Mathematische Stochastik (IMS)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/427894948?language=en
• 关键词: Smeary; Limit; Theorems; Generalized; Fr

The well known central limit theorems states that the fluctuation around their expected value of identically distributed random vectors is asymptotically normal, if rescaled with the squared root of sample size, if second moments exist. This fundamental fact is the basis of numerous inferential statistical methods, and without it, applied statistics is hardly thinkable. Driven by applications in modern pattern recognition, image processing and computational biology, the focus of statistical theory has shifted to non vector-valued data. Initially, these were random direction on the circle or the sphere (e.g. in meteorology and astronomy), random rotations (e.g. in robotics and biomechanics), or random elements in complex projective spaces (from statistical shape analysis of two-dimensional configurations).Around the turn of the millennium, employing differential geometry methods, two workings groups (Hendriks and Landsman (1998); Bhattacharya and Patrangenaru (2005)) succeeded in proving analog limit theorems on manifolds in local charts, under suitable, partially rather technical conditions. Thus, they provided, given these technical conditions, inferential methods, also for non-Euclidean data. For so-called intrinsic Fréchet means on Riemannian manifolds - these are minimizers of so-called Fréchet functions (which require existence of second moments) - there are, in principle, three such conditions:(a) uniqueness of the population mean,(b) full rank of the Hessian of the population Fréchet function at the population Fréchet mean,(c) convergence of the empirical process of the Hessian of the empirical Fréchet function indexed in a random sequence converging to the population mean.On the circle, jointly with the collaborate research partner T. Hotz (Ilmenau), in preliminary work (Hotz and Huckemann (2015)), the applicant gave examples in which under condition (a), conditions (b) and (c) fail. In consequence, in comparison to the central limit theorem, the asymptotic rate is lowered (Abbildung 1 gives the underlying intuition), giving "smeary" limit theorems. It is the aim of this research proposal, to systematically explore these novel smeary limit theorems, and building on these novel asymptotics, develop new inferential statistical methods for non-Euclidean data.

众所周知的中心极限定理指出，如果存在二阶矩，则如果用样本大小的平方根重新调整，围绕相同分布的随机向量的期望值的波动是渐近正态的，这一基本事实是许多推论统计方法的基础。没有它，应用统计学几乎是不可想象的。在现代模式识别、图像处理和计算生物学应用的推动下，统计理论的焦点已经转移到非矢量值数据上。最初，这些数据是圆或球体上的随机方向。 （例如在气象学和天文学中）、随机旋转（例如在机器人学和生物力学中）或复杂射影空间中的随机元素（来自二维配置的统计形状分析）。在世纪之交，采用微分几何方法，两个工作组（Hendriks 和 Landsman (1998)；Bhattacharya 和 Patrangaru (2005)）成功证明了模拟极限定理因此，在适当的、部分技术条件下，他们提供了对于非欧几里得数据的推理方法，即所谓的黎曼流形上的固有 Fréchet 均值。称为 Fréchet 函数（要求存在二阶矩） - 原则上存在三个这样的条件：(a) 总体均值的唯一性，(b) 总体 Fréchet 的 Hessian 矩阵的满阶总体 Fréchet 均值处的函数，(c) 以随机序列索引的经验 Fréchet 函数的 Hessian 经验过程收敛于总体均值。在圆上，与合作研究伙伴 T. Hotz (Ilmenau) 联合，在前期工作中（Hotz 和 Huckemann（2015）），申请人给出了在条件（a）下，条件（b）和（c）失败的例子，结果与中心相比。极限定理，渐进率降低（Abbildung 1给出了底层直觉），给出了“涂抹”极限定理。本研究提案的目的是系统地探索这些新颖的涂抹极限定理，并在这些新颖的渐进基础上发展。非欧几里得数据的新推论统计方法。