Functional Adaptive Huber Linear Regression

Abstract. Robust estimation has played an important role in statistical and machine learning. However, its applications to functional linear regression are still underdeveloped. In this paper, we focus on Huber’s loss with a diverging robustness parameter which was previously used in parametric models. Compared to other robust methods such as median regression, the distinction is that the proposed method aims to estimate the conditional mean robustly, instead of estimating the conditional median. We only require [Formula: see text]th moment assumption ([Formula: see text]) on the noise distribution, and the established error bounds match the optimal rate in the least-squares case as soon as [Formula: see text]. We establish convergence rate in probability when the functional predictor has a finite 4th moment, and finite-sample bound with exponential tail when the functional predictor is Gaussian, in terms of both prediction error and [Formula: see text] error. The results also extend to the case of functional estimation in a reproducing kernel Hilbert space (RKHS).