Optimal functional supervised classification with separation condition

TL;DR

This paper develops an optimal plug-in classifier for Gaussian functional data, providing tight risk bounds and demonstrating its superiority over k-nearest neighbors in functional classification tasks.

Contribution

It introduces a new optimal classifier for Gaussian functional models and establishes matching upper and lower bounds on its excess risk, advancing theoretical understanding.

Findings

Derived upper bounds on excess risk for the proposed classifier.

Proved the optimality of the classifier up to logarithmic factors.

Showed k-nearest neighbors is suboptimal in this setting.

Abstract

We consider the binary supervised classification problem with the Gaussian functional model introduced in [7]. Taking advantage of the Gaussian structure, we design a natural plug-in classifier and derive a family of upper bounds on its worst-case excess risk over Sobolev spaces. These bounds are parametrized by a separation distance quantifying the difficulty of the problem, and are proved to be optimal (up to logarithmic factors) through matching minimax lower bounds. Using the recent works of [9] and [14] we also derive a logarithmic lower bound showing that the popular k-nearest neighbors classifier is far from optimality in this specific functional setting.