Multivariate intrinsic local polynomial regression on isometric Riemannian manifolds. Applications to symmetric positive data

Introduction We extend intrinsic local polynomial regression (ILPR) to multivariate covariates on arbitrary Riemannian manifolds, providing a general framework to study the action of isometric maps on manifold-valued regression estimators. Methods We derive a closed-form expression for the ILPR estimator on manifolds endowed with an Euclidean pullback metric (EPM), a broad class that encompasses infinitely many Riemannian metrics. Pseudocode is provided along with an explicit upper bound on computational complexity. For the multivariate local linear case, we characterize the asymptotic properties of the ILPR–EPM estimator, including a closed-form expression for its bias, establishing statistical consistency under EPM. Results Simulation studies on the manifold of symmetric positive definite (SPD) matrices—benchmarking diffusion tensor imaging (DTI) data and high-dimensional covariance models—show that the Log–Cholesky metric achieves a favorable trade-off between estimation accuracy and computational efficiency. Using real neurophysiological data from the Harmonized-MultiNational quantitative electroencephalography (HarMNqEEG) dataset (N = 1,965), we apply ILPR to symmetric positive definite cross-spectral covariance matrices summarized via a 360-region cortical parcellation. Discussion We show that Log–Euclidean and Log–Cholesky estimators yield statistically and neurophysiologically indistinguishable population-level results, while the Log–Cholesky formulation substantially reduces computational cost. These results establish ILPR–Log–Cholesky as a statistically sound and computationally efficient intrinsic regression method for large-scale, high-dimensional manifold-valued data.