  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.
