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August 2020 Optimal functional supervised classification with separation condition
Sébastien Gadat, Sébastien Gerchinovitz, Clément Marteau
Bernoulli 26(3): 1797-1831 (August 2020). DOI: 10.3150/19-BEJ1170

Abstract

We consider the binary supervised classification problem with the Gaussian functional model introduced in (Math. Methods Statist. 22 (2013) 213–225). 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 (In Advances in Neural Information Processing Systems (2014) 3437–3445 Curran Associates) and (Ann. Statist. 44 (2016) 982–1009), we also derive a logarithmic lower bound showing that the popular $k$-nearest neighbors classifier is far from optimality in this specific functional setting.

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Sébastien Gadat. Sébastien Gerchinovitz. Clément Marteau. "Optimal functional supervised classification with separation condition." Bernoulli 26 (3) 1797 - 1831, August 2020. https://doi.org/10.3150/19-BEJ1170

Information

Received: 1 March 2018; Revised: 1 October 2019; Published: August 2020
First available in Project Euclid: 27 April 2020

zbMATH: 07193943
MathSciNet: MR4091092
Digital Object Identifier: 10.3150/19-BEJ1170

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 3 • August 2020
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