Functional Regression on Manifold with Contamination

TL;DR

This paper introduces a novel functional regression method for predictors on manifolds, effectively handling noise and contamination, achieving faster convergence rates than existing approaches, and demonstrating superior numerical performance.

Contribution

The paper develops a functional local linear manifold smoothing technique that adapts to manifold dimension and contamination, improving convergence rates in functional regression.

Findings

Polynomial convergence rate achieved, outperforming logarithmic rates in literature.

Phase transition observed between manifold dimension and contamination level.

Numerical experiments show superior performance over existing methods.

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.