Surface-based data is commonly observed in diverse practical applications spanning various fields. In this paper, we introduce a novel nonparametric method to discover the underlying signals from data distributed on complex surface-based domains. Our approach involves a penalized spline estimator defined on a triangulation of surface patches, which enables effective signal extraction and recovery. The proposed method offers several advantages over existing methods, including superior handling of "leakage" or "boundary effects" over complex domains, enhanced computational efficiency, and potential applications in analyzing sparse and irregularly distributed data on complex objects. We provide rigorous theoretical guarantees for the proposed method, including convergence rates of the estimator in both the $L_2$ and supremum norms, as well as the asymptotic normality of the estimator. We also demonstrate that the convergence rates achieved by our estimation method are optimal within the framework of nonparametric estimation. Furthermore, we introduce a bootstrap method to quantify the uncertainty associated with the proposed estimators accurately. The superior performance of the proposed method is demonstrated through simulation experiments and data applications on cortical surface functional magnetic resonance imaging data and oceanic near-surface atmospheric data.

翻译：表面数据广泛存在于跨多个领域的实际应用中。本文提出了一种新颖的非参数方法，用于从复杂表面域上的分布数据中提取潜在信号。我们的方法涉及定义在表面片三角剖分上的惩罚样条估计量，能够有效实现信号提取与恢复。与现有方法相比，所提方法具有多项优势，包括在复杂域上更优地处理“泄漏”或“边界效应”、更高的计算效率，以及分析复杂物体上稀疏不规则分布数据的潜在应用。我们为所提方法提供了严格的理论保证，包括估计量在$L_2$范数和上确界范数下的收敛速度以及估计量的渐近正态性。同时证明了该估计方法在非参数估计框架下达到了最优收敛速度。此外，我们引入了一种自助法来准确量化所提估计量的不确定性。通过模拟实验以及在皮层表面功能性磁共振成像数据和海洋近地表大气数据上的应用，验证了所提方法的优越性能。