Optimal functional supervised classification with separation condition

Sébastien Gadat (1) , Sébastien Gerchinovitz (2) , Clément Marteau (3)

1 TSE-R - Toulouse School of Economics
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 PSPM - Probabilités, statistique, physique mathématique

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.

Domaines: Théorie [stat.TH]

Sébastien Gadat, Sébastien Gerchinovitz, Clément Marteau. Optimal functional supervised classification with separation condition. 2018. ⟨hal-01679648v1⟩