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.

翻译：本文提出了一种基于等距黎曼流形（IRMs）的新型非参数黎曼回归方法。所提出的技术——IRMs上的本征局部多项式回归（ILPR-IRMs）——能够在保持底层几何结构的同时实现黎曼流形间的全局数据映射。ILPR方法被推广至处理任意黎曼流形及等距变换上的多元协变量。具体而言，对于配备欧几里得拉回度量（EPMs）的流形，我们推导出了多元ILPR（ILPR-EPM）的闭式解析公式。建立了多元局部线性情形下ILPR-EPM的渐近统计性质，包括渐近偏差公式，从而证明了估计量的一致性。本文通过聚焦于机器学习和神经科学中出现的对称正定（SPD）流形上的一组黎曼度量，展示了该方法的可能应用。我们证明了SPD流形上的若干度量均为EPMs，从而得到了SPD流形上多元ILPR估计量的闭式解析表达式。本文在SPD流形模拟数据上比较了两种特定EPM（对数-乔列斯基度量与对数-欧几里得度量）下ILPR估计量的性能，并通过缩放流形与协变量维度时的仿射不变度量将其与外蕴LPR进行对比。结果表明，采用对数-乔列斯基度量的ILPR计算速度更快，且在误差与时间之间提供了更优的权衡。最后，我们利用SPD流形上的对数-乔列斯基度量实现了用于高维SPD数据可视化的高效本征型Rie-SNE算法。用于实现ILPR-EPMs及其他相关计算的代码已在GitHub页面公开。