Multivariate Intrinsic Local Polynomial Regression on Isometric Riemannian Manifolds: Applications to Positive Definite Data

The paper introduces a novel non-parametric Riemannian regression method using Isometric Riemannian Manifolds (IRMs). The proposed technique, Intrinsic Local Polynomial Regression on IRMs (ILPR-IRMs), enables global data mapping between Riemannian manifolds while preserving underlying geometries. The ILPR method is generalized to handle multivariate covariates on any Riemannian manifold and isometry. Specifically, for manifolds equipped with Euclidean Pullback Metrics (EPMs), a closed analytical formula is derived for the multivariate ILPR (ILPR-EPM). Asymptotic statistical properties of the ILPR-EPM for the multivariate local linear case are established, including a formula for the asymptotic bias, establishing estimator consistency. The paper showcases possible applications of the method by focusing on a group of Riemannian metrics on the Symmetric Positive Definite (SPD) manifold, which arises in machine learning and neuroscience. It is demonstrated that several metrics on the SPD manifold are EPMs, resulting in a closed analytical expression for the multivariate ILPR estimator on the SPD manifold. The paper evaluates the ILPR estimator's performance under two specific EPMs, Log-Cholesky and Log-Euclidean, on simulated data on the SPD manifold and compares it with extrinsic LPR using the Affine-Invariant when scaling the manifold and covariate dimension. The results show that the ILPR using the Log-Cholesky metric is computationally faster and provides a better trade-off between error and time than other metrics. Finally, the Log-Cholesky metric on the SPD manifold is employed to implement an efficient and intrinsic version of Rie-SNE for visualizing high-dimensional SPD data. The code for implementing ILPR-EPMs and other relevant calculations is available on the GitHub page.