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表明，如果用样本量的平方根重新缩放，如果存在秒矩，则同分布随机向量在期望值周围的波动是渐近正态的。这个基本事实是许多推理统计方法的基础，没有它，应用统计几乎是不可想象的。在现代模式识别、图像处理和计算生物学应用的推动下，统计理论的重点已经转移到非向量值数据上。最初，这些是圆或球上的随机方向（例如在气象学和天文学中），随机旋转（例如在机器人和生物力学中），或复杂投影空间中的随机元素（来自二维构型的统计形状分析）。在世纪之交，采用微分几何方法，两个工作小组(Hendriks和Landsman (1998)；Bhattacharya和Patrangenaru(2005))成功地证明了局部图中流形的模拟极限定理，在适当的，部分相当技术性的条件下。因此，在这些技术条件下，他们也提供了非欧几里得数据的推理方法。对于黎曼流形上所谓的内在fracimchet means——这些是所谓fracimchet函数的最小值（要求存在第二矩）——原则上有三个这样的条件：(a)总体均值的唯一性，(b)总体fracimchet函数的Hessian在总体fracimchet均值处的满秩，(c)经验fracimchet函数的Hessian的经验过程收敛于收敛于总体均值的随机序列。在圈子上，申请人与合作研究伙伴T. Hotz （Ilmenau）共同，在初步工作（Hotz and Huckemann(2015)）中，给出了条件(a)，条件(b)和条件(c)失败的例子。因此，与中心极限定理相比，渐近率降低了（Abbildung 1给出了潜在的直觉），给出了“模糊的”极限定理。本研究计划的目的是系统地探索这些新的smeary极限定理，并在这些新的渐近性的基础上，开发新的非欧几里得数据的推理统计方法。