Intrinsic Footpoint-invariant Riemannian Cross-covariance

2026-06-08

AI Summary

Traditional covariance struggles to effectively model second-order statistics on nonlinear Riemannian manifolds such as the space of symmetric positive-definite (SPD) matrices and Kendall’s shape space. This work proposes the first intrinsic and basepoint-invariant definition of Riemannian cross-covariance, achieved by parallel transporting local variations on the manifold to a common tangent space. The resulting estimator inherits desirable properties of Euclidean covariance and is supported by rigorous asymptotic theory. Efficient computational procedures are developed for spheres, SPD manifolds, and Kendall shape spaces. Numerical experiments and real-world analysis of heart valve shapes demonstrate the method’s effectiveness, establishing a foundational second-order statistical tool for representation learning with non-Euclidean data.

Abstract

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.

Problem

• Riemannian manifold
• covariance estimation
• non-Euclidean data
• SPD matrices
• shape analysis

Innovation

• Riemannian covariance
• parallel transport
• manifold-valued data
• footpoint invariance
• second-order statistics