Covariance estimation yields a fundamental second-order statistic underlying representation learning, dimension reduction, and dependence modeling. While covariance has been well understood in Euclidean spaces, it is ill-defined for random objects residing on nonlinear Riemannian manifolds, which increasingly arise in modern machine learning applications involving shapes, symmetric positive definite (SPD) matrices, etc. This paper introduces an intrinsic Riemannian cross-covariance for manifold-valued random objects. Our approach defines covariance and correlation by transporting local variations to a common tangent space via parallel transport, yielding a second-order descriptor that is independent of arbitrary coordinate choices. We establish that the proposed covariance inherits desirable properties of its Euclidean counterparts and characterize its asymptotic behavior. Numerical studies on spheres and SPD manifolds, together with real-data experiments on heart valve shapes in Kendall's shape space, demonstrate the effectiveness of our estimators and verify the stated properties. Our results position the Riemannian covariance as a fundamental tool for second-order learning and analysis in non-Euclidean representation spaces.

翻译：协方差估计是表示学习、降维和依赖建模中基础的二阶统计量。尽管协方差在欧几里得空间中已得到充分理解，但对于存在于非线性黎曼流形上的随机对象（例如现代机器学习应用中日益常见的形状、对称正定（SPD）矩阵等），其定义却不明确。本文引入了一种针对流形值随机对象的内在黎曼交叉协方差。我们的方法通过平行移动将局部变化传输至公共切空间来定义协方差和相关关系，从而得到与任意坐标选择无关的二阶描述子。我们证明所提出的协方差继承了其欧几里得对应物的理想性质，并刻画了其渐近行为。在球面和SPD流形上的数值研究，以及基于肯德尔形状空间中心脏瓣膜形状的真实数据实验，验证了我们估计量的有效性及其所述性质。我们的研究结果将黎曼协方差定位为非欧几里得表示空间中二阶学习与分析的基础工具。