We study extensions of Fr\'{e}chet means for random objects in the space ${\rm Sym}^+(p)$ of $p \times p$ symmetric positive-definite matrices using the scaling-rotation geometric framework introduced by Jung et al. [\textit{SIAM J. Matrix. Anal. Appl.} \textbf{36} (2015) 1180-1201]. The scaling-rotation framework is designed to enjoy a clearer interpretation of the changes in random ellipsoids in terms of scaling and rotation. In this work, we formally define the \emph{scaling-rotation (SR) mean set} to be the set of Fr\'{e}chet means in ${\rm Sym}^+(p)$ with respect to the scaling-rotation distance. Since computing such means requires a difficult optimization, we also define the \emph{partial scaling-rotation (PSR) mean set} lying on the space of eigen-decompositions as a proxy for the SR mean set. The PSR mean set is easier to compute and its projection to ${\rm Sym}^+(p)$ often coincides with SR mean set. Minimal conditions are required to ensure that the mean sets are non-empty. Because eigen-decompositions are never unique, neither are PSR means, but we give sufficient conditions for the sample PSR mean to be unique up to the action of a certain finite group. We also establish strong consistency of the sample PSR means as estimators of the population PSR mean set, and a central limit theorem. In an application to multivariate tensor-based morphometry, we demonstrate that a two-group test using the proposed PSR means can have greater power than the two-group test using the usual affine-invariant geometric framework for symmetric positive-definite matrices.

翻译：我们研究了随机对象在p×p对称正定矩阵空间${\rm Sym}^+(p)$中的Fr\'{e}chet均值推广问题，基于Jung等人[《SIAM J. Matrix. Anal. Appl.》\textbf{36} (2015) 1180-1201]提出的缩放-旋转几何框架。该框架旨在通过缩放和旋转对随机椭球的变化提供更清晰的解释。本文正式定义\textit{缩放-旋转（SR）均值集}为相对于缩放-旋转距离在${\rm Sym}^+(p)$中的Fr\'{e}chet均值集合。由于计算此类均值需要困难的最优化过程，我们同时定义位于特征分解空间上的\textit{部分缩放-旋转（PSR）均值集}作为SR均值集的代理。PSR均值集更易计算，且其到${\rm Sym}^+(p)$的投影通常与SR均值集一致。确保均值集非空仅需极少的条件。由于特征分解不唯一，PSR均值也不唯一，但我们给出充分条件使得样本PSR均值在特定有限群作用下唯一。我们还建立了样本PSR均值作为总体PSR均值集估计量的强相合性及中心极限定理。在多变量张量形态测量学的应用中，我们证明基于所提PSR均值的两组检验比基于传统仿射不变几何框架的对称正定矩阵两组检验具有更高的统计功效。