Optimal Classification for Functional Data

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.