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Intrinsic Riemannian Cross-covariance for Manifold-valued Random Objects

NeurIPS 2026 · Learning theory and algorithms › Matrix and tensor methods · approach: Geometry of low-rank and SPD structure · 0 citations (OpenAlex)

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Proposes an intrinsic, footpoint-invariant Riemannian cross-covariance and correlation for manifold-valued random variables, using parallel transport to a common tangent space, with consistency and asymptotic normality results and experiments on spheres, SPD matrices, and Kendall shape space.

Problem

Covariance and correlation are undefined for random objects on nonlinear Riemannian manifolds because tangent spaces at different points cannot be compared; prior two-variable definitions depend on footpoint choice and need shared support.

Approach

Transport the log-maps of each variable (taken at its Fréchet mean) to a common tangent space at a point on the geodesic joining the two means via parallel transport, then take the trace of the centered cross second moment; the resulting trace is invariant to footpoint along that geodesic and normalized to [-1,1].

Key results

• On the sphere, the proposed correlation is numerically constant along the geodesic between Fréchet means, whereas the A&P correlation shows an inverted-U pattern peaking at the joint Fréchet mean.
• On SPD matrices, the proposed estimator has SE about 0.008 across 21 footpoints versus about 0.15 for A&P.
• On the Sunnybrook LV data, correlations were 0.587 (Normal), 0.352 (Hypertrophy), 0.840 (HF-infarct), 0.628 (HF+infarct), close to A&P values, while CCA gave 1 for all groups.

Surprising findings

• Standard CCA returns correlation exactly 1 in all four small-sample groups, attributed to overfitting.

Limitations

• Requires unique Fréchet means and d(mu,nu) below injectivity radius, plus compact/bounded support assumptions.
• Parallel transport is only well-defined along the geodesic joining the means; Kendall shape space uses approximate pole-ladder transport.
• Footpoint invariance proof is stated for points on the mean geodesic; the scaling operation is defined ad hoc.

Experimental setup

Datasets: Sunnybrook Cardiac MRI dataset

Benchmarks: Sunnybrook Cardiac MRI heart valve/LV shape groups (Kendall shape space)

Metrics: Riemannian correlation estimate covariance variance footpoint invariance (SE along geodesic) confidence intervals

Baselines: Abuqrais-Pigoli (A&P) correlation canonical correlation analysis (CCA)

Builds on: Abuqrais-Pigoli Riemannian covariance Pennec covariance Wasserstein covariance Riemannian CCA

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