发表机构
Indian Statistical Institute; VSB Technical University of Ostrava（印度统计研究所; 俄斯特拉发技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究流形上束值统计量，通过希尔伯特空间不等式推导集中界，分离偏差与方差，给出精确公式，建立稳健估计器，经实验验证理论预测。
AI 中文摘要
许多几何统计和流形学习管道通常产生的观测值，如切向量或局部标架，其天然所在是附着于基流形不同点的变化纤维族，而非单个共享向量空间。形成经验平均值需要将这些观测值传输到共同参考纤维，从而引入经典集中理论中不存在的曲率和全纯性驱动效应。我们为此类传输的经验均值发展了非渐近集中理论，通过精确的希尔伯特空间不等式推导有限样本、无维数的霍夫丁型和伯恩斯坦型界。当到参考点的最短路径不唯一时，传输变得依赖路径并引入确定性全纯性偏差；我们通过束曲率和环路几何分离并量化此偏差，给出圆球切丛的精确闭式公式。所得偏差 - 方差分解将样本量\(n\)中以经典\(n^{-1/2}\)速率衰减的随机波动与曲率驱动的误差下限分开，极小极大下界证实这两项都是不可避免的。我们还建立了一个稳健的均值中位数估计器，在重尾和参考纤维中的中心极限定理下实现最优速率。在球面上的对照实验验证了所有理论预测。
英文摘要
Many geometric statistics and manifold learning pipelines routinely produce observations -- such as tangent vectors or local frames -- whose natural home is a varying family of fibers attached to different points of a base manifold, rather than a single shared vector space. Forming empirical averages requires transporting these observations to a common reference fiber, thereby introducing curvature- and holonomy-driven effects that are absent from classical concentration theory. We develop a non-asymptotic concentration theory for such transported empirical means, deriving finite-sample, dimension-free Hoeffding- and Bernstein-type bounds via sharp Hilbert-space inequalities. When shortest paths to the reference point are non-unique, transport becomes path-dependent and introduces a deterministic holonomy bias; we isolate and quantify this bias through bundle curvature and loop geometry, with sharp closed-form formulas for the tangent bundle of a round sphere. The resulting bias-variance decomposition separates the stochastic fluctuation decaying at the classical $n^{-1/2}$ rate in sample size $n$, from a curvature-driven error floor that no amount of additional data can eliminate; minimax lower bounds confirm both terms are unavoidable. We further establish a robust median-of-means estimator achieving optimal rates under heavy tails and the central limit theorem in the reference fiber. Controlled experiments on the sphere validate all theoretical predictions.
CommentsAccepted in ICML 2026
