Abstract: A central topic in functional data analysis is how to design an optimal decision rule, based on training samples, to classify a data function. We exploit the optimal classification problem in which the data functions are Gaussian processes. We derive sharp convergence rates for the minimax excess misclassification risk both when the data functions are fully observed and when they are discretely observed. We explore two easily implementable classifiers, based on a discriminant analysis and on a deep neural network, respectively, which both achieve optimality in Gaussian settings. Our deep neural network classifier is new in the literature, and demonstrates outstanding performance, even when the data functions are nonGaussian. For discretely observed data, we discover a novel critical sampling frequency that governs the sharp convergence rates. The proposed classifiers perform favorably in finite-sample applications, shown in comparisons with otherfunctional classifiers in simulations and one real-data application.

Key words and phrases: Functional classification, functional deep neural network, functional quadratic discriminant analysis, Gaussian process, minimax excess misclassification risk.