# Intrinsic Riemannian Cross-covariance for Manifold-valued Random Objects

arXiv: https://arxiv.org/abs/2606.10212 · HTML: https://mlsanity.com/paper/2606.10212/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 29 references that are in the map.
- First posted on arXiv 8 Jun 2026.

## Builds on (most cited first, 7 of 7)

- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Analyzing Dynamical Brain Functional Connectivity As Trajectories on Space of Covariance Matrices](https://mlsanity.com/paper/1904.05449/) (2019)
- [Intrinsic Riemannian Functional Data Analysis](https://mlsanity.com/paper/1812.01831/) (2018)
- [Intrinsic Riemannian Functional Data Analysis for Sparse Longitudinal Observations](https://mlsanity.com/paper/2009.07427/) (2020)
- [Clustering Tree-structured Data on Manifold](https://mlsanity.com/paper/1507.05532/) (2015)
- [Wasserstein Covariance for Multiple Random Densities](https://mlsanity.com/paper/1812.07694/) (2018)

## Built on by (most cited first, 1 of 1)

- [A Riemannian View on Active Subspaces](https://mlsanity.com/paper/2607.25163/) (2026)

## Abstract (arXiv)

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.