提出新型内蕴高斯过程回归，处理流形上响应变量的建模问题。
Intrinsic Gaussian Process Regression Modeling for Manifold-valued Response Variable
- 基于黎曼流形上的平行移动构造内蕴协方差结构
- 模型适用于无自然嵌入空间的流形数据，且具有后验一致性
- 适合流形数据分析、生物形态建模等需要几何精确性的场景
外部高斯过程回归方法（如包裹高斯过程）已被用于分析流形数据，但缺乏针对流形值响应变量的内蕴高斯过程方法。本文首次利用黎曼流形上的平行移动算子，提出一种内蕴协方差结构，解决了构建良好定义的高斯过程回归模型的关键问题。进而提出一种新的内蕴高斯过程回归模型，可应用于不仅限于欧氏子流形，也包括无自然嵌入空间的流形数据。我们建立了所提模型的渐近性质，包括信息一致性与后验一致性，并证明回归函数的后验分布对协方差函数坐标表示的正交基选择保持不变。数值实验（含模拟与真实案例）表明该方法表现良好。
原文摘要 · Abstract (English)
Extrinsic Gaussian process regression methods, such as wrapped Gaussian process, have been developed to analyze manifold data. However, there is a lack of intrinsic Gaussian process methods for studying complex data with manifold-valued response variables. In this paper, we first apply the parallel transport operator on Riemannian manifold to propose an intrinsic covariance structure that addresses a critical aspect of constructing a well-defined Gaussian process regression model. We then propose a novel intrinsic Gaussian process regression model for manifold-valued data, which can be applied to data situated not only on Euclidean submanifolds but also on manifolds without a natural ambient space. We establish the asymptotic properties of the proposed models, including information consistency and posterior consistency, and we also show that the posterior distribution of the regression function is invariant to the choice of orthonormal frames for the coordinate representations of the covariance function. Numerical studies, including simulation and real examples, indicate that the proposed methods work well.
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