Optimal classification and nonparametric regression for functional data

TL;DR

This paper derives minimax convergence rates for functional data classification and nonparametric regression, showing that conventional kernel methods can achieve optimal rates without knowing the smoothness of the target functions.

Contribution

It establishes the first minimax convergence rates for classification and regression with functional data in a general metric space, using kernel procedures without smoothness knowledge.

Findings

Optimal logarithmic convergence rates under smoothness constraints

Kernel procedures achieve these rates without prior smoothness knowledge

Applicable to functional data in general Polish metric spaces

Abstract

We establish minimax convergence rates for classification of functional data and for nonparametric regression with functional design variables. The optimal rates are of logarithmic type under smoothness constraints on the functional density and the regression mapping, respectively. These asymptotic properties are attainable by conventional kernel procedures. The bandwidth selector does not require knowledge of the smoothness level of the target mapping. In this work, the functional data are considered as realisations of random variables which take their values in a general Polish metric space. We impose certain metric entropy constraints on this space; but no algebraic properties are required.