%0 Journal Article
%@ 1350-7265
%A Gadat, Sébastien
%A Gerchinovitz, Sebastien
%A Marteau, Clément
%C Londres
%D 2020
%F publications:32803
%I International Statistical Institute
%J Bernoulli journal
%N n°3
%P 1797-1831
%R 10.3150/19-BEJ1170
%T Optimal functional supervised classification with separation condition
%U https://publications.ut-capitole.fr/id/eprint/32803/
%V vol. 26
%X 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.