In this paper, we study linear regression applied to data structured on a manifold. We assume that the data manifold is smooth and is embedded in a Euclidean space, and our objective is to reveal the impact of the data manifold's extrinsic geometry on the regression. Specifically, we analyze the impact of the manifold's curvatures (or higher order nonlinearity in the parameterization when the curvatures are locally zero) on the uniqueness of the regression solution. Our findings suggest that the corresponding linear regression does not have a unique solution when the embedded submanifold is flat in some dimensions. Otherwise, the manifold's curvature (or higher order nonlinearity in the embedding) may contribute significantly, particularly in the solution associated with the normal directions of the manifold. Our findings thus reveal the role of data manifold geometry in ensuring the stability of regression models for out-of-distribution inferences.

翻译：本文研究了应用于流形结构数据的线性回归问题。我们假设数据流形光滑且嵌入欧几里得空间，旨在揭示数据流形外在几何对回归的影响。具体而言，我们分析了流形的曲率（或参数化中曲率局部为零时的高阶非线性）对回归解唯一性的影响。研究结果表明：当嵌入子流形在某些维度上平坦时，对应的线性回归不存在唯一解；反之，流形的曲率（或嵌入中的高阶非线性）可能产生显著影响，尤其体现在与流形法向相关的解中。因此，本研究揭示了数据流形几何在确保回归模型对分布外推断稳定性方面的作用。