Optimal Classification for Functional Data

TL;DR

This paper develops optimal classifiers for functional data, including a novel deep neural network approach that performs well even on non-Gaussian data, with theoretical guarantees and practical validation.

Contribution

It introduces a new deep neural network classifier for functional data, achieving optimality in Gaussian cases and strong performance in non-Gaussian scenarios, with theoretical convergence rates.

Findings

Deep neural network classifier outperforms existing methods.

Sharp convergence rates depend on sampling frequency.

Proposed classifiers perform well in simulations and real data.

Abstract

A central topic in functional data analysis is how to design an optimaldecision rule, based on training samples, to classify a data function. We exploit the optimal classification problem when data functions are Gaussian processes. Sharp nonasymptotic convergence rates for minimax excess mis-classification risk are derived in both settings that data functions are fully observed and discretely observed. We explore two easily implementable classifiers based on discriminant analysis and deep neural network, respectively, which are both proven to achieve optimality in Gaussian setting. Our deepneural network classifier is new in literature which demonstrates outstanding performance even when data functions are non-Gaussian. In case of discretely observed data, we discover a novel critical sampling frequency thatgoverns the sharp convergence rates. The proposed classifiers perform favorably in finite-sample applications, as we demonstrate through comparisonswith other functional classifiers in simulations and one real data application.