Optimal functional supervised classification with separation condition

Sébastien Gadat (sebastien.gadat@math.univ-toulouse.fr), Sebastien Gerchinovitz and Clément Marteau

No 18-904, TSE Working Papers from Toulouse School of Economics (TSE)

Abstract: 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.

Date: 2018-03
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