Let $(M,g)$ be a Riemannian manifold. If $\mu$ is a probability measure on $M$ given by a continuous density function, one would expect the Fr\'{e}chet means of data-samples $Q=(q_1,q_2,\dots, q_N)\in M^N$, with respect to $\mu$, to behave ``generically''; e.g. the probability that the Fr\'{e}chet mean set $\mbox{FM}(Q)$ has any elements that lie in a given, positive-codimension submanifold, should be zero for any $N\geq 1$. Even this simplest instance of genericity does not seem to have been proven in the literature, except in special cases. The main result of this paper is a general, and stronger, genericity property: given i.i.d. absolutely continuous $M$-valued random variables $X_1,\dots, X_N$, and a subset $A\subset M$ of volume-measure zero, $\mbox{Pr}\left\{\mbox{FM}(\{X_1,\dots,X_N\})\subset M\backslash A\right\}=1.$ We also establish a companion theorem for equivariant Fr\'{e}chet means, defined when $(M,g)$ arises as the quotient of a Riemannian manifold $(\widetilde{M},\tilde{g})$ by a free, isometric action of a finite group. The equivariant Fr\'{e}chet means lie in $\widetilde{M}$, but, as we show, project down to the ordinary Fr\'{e}chet sample means, and enjoy a similar genericity property. Both these theorems are proven as consequences of a purely geometric (and quite general) result that constitutes the core mathematics in this paper: If $A\subset M$ has volume zero in $M$ , then the set $\{Q\in M^N : \mbox{FM}(Q) \cap A\neq\emptyset\}$ has volume zero in $M^N$. We conclude the paper with an application to partial scaling-rotation means, a type of mean for symmetric positive-definite matrices.

翻译：设$(M,g)$为黎曼流形。若$\mu$是由连续密度函数定义的$M$上的概率测度，则关于$\mu$的数据样本$Q=(q_1,q_2,\dots, q_N)\in M^N$的Fr\'{e}chet均值应具有“泛型”行为；例如，对任意$N\geq 1$，Fr\'{e}chet均值集$\mbox{FM}(Q)$中存在元素位于给定正余维子流形上的概率应为零。即使这一最简泛型实例，除特殊情况外，文献中似乎尚未证明。本文的主要结果是如下更一般的强泛型性质：给定独立同分布的绝对连续$M$值随机变量$X_1,\dots, X_N$，以及体积测度为零的子集$A\subset M$，有$\mbox{Pr}\left\{\mbox{FM}(\{X_1,\dots,X_N\})\subset M\backslash A\right\}=1.$ 我们同时建立了等变Fr\'{e}chet均值的伴随定理，该均值定义于$(M,g)$作为黎曼流形$(\widetilde{M},\tilde{g})$关于有限群自由等距作用的商空间的情形。等变Fr\'{e}chet均值位于$\widetilde{M}$中，但如我们所示，它们可投影到普通Fr\'{e}chet样本均值，并具有类似泛型性质。这两个定理均作为本文核心数学内容的纯几何（且相当一般）结论的推论而证明：若$A\subset M$在$M$中体积为零，则集合$\{Q\in M^N : \mbox{FM}(Q) \cap A\neq\emptyset\}$在$M^N$中体积为零。我们以偏缩放-旋转均值（一种对称正定矩阵的均值形式）的应用作为论文结语。