Abstract: We investigate perfect classification on functional data using finite samples. Perfect classification for functional data is easier to achieve than for finite-dimensional data, because a sufficient condition for the existence of a perfect classifier, called the Delaigle-Hall condition, is available only for functional data. However, a large sample size is required to achieve perfect classification, even when the Delaigle-Hall condition holds, because the minimax convergence rate of the errors with functional data has a logarithm order in the sample size. We resolve this complication by proving that the Delaigle-Hall condition also achieves fast convergence of the misclassification error in a sample size under the bounded entropy condition on functional data. We study a reproducing kernel Hilbert space-based classifier under the Delaigle-Hall condition, and show that the convergence rate of its misclassification error has an exponential order in the sample size. Technically, our proof is based on (i) connecting the Delaigle-Hall condition and a margin of classifiers, and (ii) handling metric entropy of functional data. The results of our experiments support our findings, and show that other classifiers for functional data have a similar property.

Key words and phrases: Convergence rate, functional data, perfect classification, reproducing kernel Hilbert space.