[论文解读] Curvature effects on the empirical mean in Riemannian and affine Manifolds: a non-asymptotic high concentration expansion in the small-sample regime

Xavier Pennec|arXiv (Cornell University)|Jun 18, 2019

Morphological variations and asymmetry参考文献 24被引用 13

一句话总结

本文推导了黎曼流形和仿射联络流形上经验弗雷歇均值的非渐近高斯集中展开式，表明曲率在均值中引入了 $1/n$ 阶偏差，并通过曲率张量调节协方差收敛速率。关键结果是一个标量乘法因子，编码了负曲率如何加速收敛（快于欧氏空间），而正曲率则使其变慢，且在Karcher-Kendall集中阈值处发散。

ABSTRACT

The asymptotic concentration of the Fr{é}chet mean of IID random variables on a Rieman-nian manifold was established with a central limit theorem by Bhattacharya \& Patrangenaru (BP-CLT) [6]. This asymptotic result shows that the Fr{é}chet mean behaves almost as the usual Euclidean case for sufficiently concentrated distributions. However, the asymptotic covariance matrix of the empirical mean is modified by the expected Hessian of the squared distance. This Hessian matrix was explicitly computed in [5] for constant curvature spaces in order to relate it to the sectional curvature. Although explicit, the formula remains quite difficult to interpret, and the intuitive effect of the curvature on the asymptotic convergence remains unclear. Moreover, we are most often interested in the mean of a finite sample of small size in practice. In this work, we aim at understanding the effect of the manifold curvature in this small sample regime. Last but not least, one would like computable and interpretable approximations that can be extended from the empirical Fr{é}chet mean in Rie-mannian manifolds to the empirical exponential barycenters in affine connection manifolds. For distributions that are highly concentrated around their mean, and for any finite number of samples, we establish explicit Taylor expansions on the first and second moment of the empirical mean thanks to a new Taylor expansion of the Riemannian log-map in affine connection spaces. This shows that the empirical mean has a bias in 1/n proportional to the gradient of the curvature tensor contracted twice with the covariance matrix, and a modulation of the convergence rate of the covariance matrix proportional to the covariance-curvature tensor. We show that our non-asymptotic high concentration expansion is consistent with the asymptotic expansion of the BP-CLT. Experiments on constant curvature spaces demonstrate that both expansions are very accurate in their domain of validity. Moreover, the modulation of the convergence rate of the empirical mean's covariance matrix is explicitly encoded using a scalar multiplicative factor that gives an intuitive vision of the impact of the curvature: the variance of the empirical mean decreases faster than in the Euclidean case in negatively curved space forms, with an infinite speed for an infinite negative curvature. This suggests potential links with the stickiness of the Fr{é}chet mean described in stratified spaces. On the contrary, the variance of the empirical mean decreases more slowly than in the Euclidean case in positive curvature space forms, with divergence when we approach the limits of the Karcher \& Kendall concentration conditions with a uniform distribution on the equator of the sphere, for which the Fr{é}chet mean is not a single point any more.

研究动机与目标

• 理解在小样本情形下曲率对经验弗雷歇均值的影响，此时渐近近似失效。
• 提供超越渐近理论的、可计算且可解释的经验均值一阶与二阶矩的近似表达式。
• 通过指数重心将结果从黎曼流形推广至仿射联络流形。
• 阐明曲率对均值估计速度与偏差的直观影响，尤其是在高度集中分布的情况下。
• 在高斯集中极限下，将非渐近展开式与Bhattacharya-Patrangenaru中心极限定理相协调。

提出的方法

• 利用Gavrilov公式推导黎曼对数映射的坐标无关泰勒展开，其仅依赖于曲率张量与挠率张量及其协变导数。
• 将此展开式应用于计算小样本情形下经验均值一阶与二阶矩的显式高阶近似。
• 提出一种新表述方式，通过使用张量形式的内在表达式而非基于坐标的展开，避免了复杂的指标缩并。
• 利用协方差-曲率张量量化曲率如何调节经验均值协方差矩阵的收敛速率。
• 在高斯集中、大样本极限下，验证其与Bhattacharya-Patrangenaru中心极限定理的一致性。
• 在常曲率空间上验证该展开式，显示其与已知渐近结果一致，并提供了直观的标量调制因子。

实验结果

研究问题

• RQ1 在有限样本设置下，曲率如何影响经验弗雷歇均值的偏差？是否能通过非渐近展开式捕捉到这一效应？
• RQ2 黎曼曲率张量在调节经验均值协方差矩阵收敛速率方面的确切作用是什么？
• RQ3 曲率对经验均值的影响能否通过一个简单标量因子来表达，以捕捉其相对于欧氏情形的收敛加速或减速？
• RQ4 在高斯集中情形下，非渐近展开式与渐近Bhattacharya-Patrangenaru CLT相比如何？
• RQ5 该框架能否从黎曼流形推广至仿射联络流形，以适用于指数重心？

主要发现

• 经验弗雷歇均值表现出与 $1/n$ 阶成比例的偏差，其比例系数为曲率张量梯度与协方差矩阵两次缩并的结果。
• 经验均值协方差矩阵的收敛速率受协方差-曲率张量调制，该张量引入了一个依赖于曲率的标量乘法因子。
• 在负曲率空间形式中，经验均值的方差比欧氏情形减小得更快，当曲率趋于负无穷时趋近于零。
• 在正曲率空间形式中，方差比欧氏情形减小得更慢，当趋近Karcher-Kendall集中条件时发散，例如在球面上具有均匀赤道分布的情形。
• 非渐近展开式在高斯集中极限下与Bhattacharya-Patrangenaru中心极限定理一致，验证了其准确性。
• 标量调制因子提供了直观的几何解释：负曲率增强集中性，而正曲率则阻碍集中性，可能与分层空间中的粘滞现象相关。