The extension of bivariate measures of dependence to non-Euclidean spaces is a challenging problem. The non-linear nature of these spaces makes the generalisation of classical measures of linear dependence (such as the covariance) not trivial. In this paper, we propose a novel approach to measure stochastic dependence between two random variables taking values in a Riemannian manifold, with the aim of both generalising the classical concepts of covariance and correlation and building a connection to Fr\'echet moments of random variables on manifolds. We introduce generalised local measures of covariance and correlation and we show that the latter is a natural extension of Pearson correlation. We then propose suitable estimators for these quantities and we prove strong consistency results. Finally, we demonstrate their effectiveness through simulated examples and a real-world application.

翻译：将双变量依赖度量推广到非欧几里得空间是一个具有挑战性的问题。这些空间的非线性特性使得经典线性依赖度量（如协方差）的推广并非易事。本文提出了一种新颖的方法，用于度量两个在黎曼流形上取值的随机变量之间的随机依赖性，旨在同时推广经典的协方差和相关性概念，并建立与流形上随机变量的Fr\'echet矩之间的联系。我们引入了广义的局部协方差和相关性度量，并证明了后者是皮尔逊相关性的自然推广。随后，我们为这些量提出了合适的估计量，并证明了强一致性结果。最后，我们通过模拟示例和一个实际应用证明了其有效性。