Regression on manifolds, and, more broadly, statistics on manifolds, has garnered significant importance in recent years due to the vast number of applications for this type of data. Circular data is a classic example, but so is data in the space of covariance matrices, data on the Grassmannian manifold obtained as a result of principal component analysis, among many others. In this work we investigate prediction sets for regression scenarios when the response variable, denoted by $Y$, resides in a manifold, and the covariable, denoted by X, lies in Euclidean space. This extends the concepts delineated in [Lei and Wasserman, 2014] to this novel context. Aligning with traditional principles in conformal inference, these prediction sets are distribution-free, indicating that no specific assumptions are imposed on the joint distribution of $(X, Y)$, and they maintain a non-parametric character. We prove the asymptotic almost sure convergence of the empirical version of these regions on the manifold to their population counterparts. The efficiency of this method is shown through a comprehensive simulation study and an analysis involving real-world data.

翻译：流形回归，以及更广义的流形统计学，近年来因该类数据的广泛应用而变得愈发重要。圆形数据是经典例子，但协方差矩阵空间中的数据、通过主成分分析获得的格拉斯曼流形上的数据等也属此类。本文研究响应变量$Y$位于流形、协变量$X$位于欧氏空间时回归场景下的预测集。这将在[Lei and Wasserman, 2014]中阐述的概念扩展至这一新情境。与共形推断的传统原理一致，这些预测集是分布自由的，即不对$(X, Y)$的联合分布施加特定假设，且保持非参数特性。我们证明了这些流形上区域的经验版本几乎必然渐近收敛于其总体版本。通过全面的模拟研究和真实数据分析，展示了该方法的有效性。