We propose a new method for functional nonparametric regression with a predictor that resides on a finite-dimensional manifold but is only observable in an infinite-dimensional space. Contamination of the predictor due to discrete/noisy measurements is also accounted for. By using functional local linear manifold smoothing, the proposed estimator enjoys a polynomial rate of convergence that adapts to the intrinsic manifold dimension and the contamination level. This is in contrast to the logarithmic convergence rate in the literature of functional nonparametric regression. We also observe a phase transition phenomenon regarding the interplay of the manifold dimension and the contamination level. We demonstrate that the proposed method has favorable numerical performance relative to commonly used methods via simulated and real data examples.

中文翻译：

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带有污染的歧管的功能回归

我们提出了一种新的函数非参数回归方法，其预测器位于有限维流形上，但只能在无限维空间中观察到。由于离散/噪声测量而导致的预测器污染也被考虑在内。通过使用函数局部线性流形平滑，所提出的估计器享有多项式收敛速度，该收敛速度适应内在流形维度和污染水平。这与函数非参数回归文献中的对数收敛速度形成对比。我们还观察到有关歧管尺寸和污染水平相互作用的相变现象。我们通过模拟和真实数据示例证明了所提出的方法相对于常用方法具有良好的数值性能。