Functional Regression on Manifold with Contamination

Abstract: 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.