Minimax fast rates for discriminant analysis with errors in variables

TL;DR

This paper investigates the impact of measurement errors in discriminant analysis, establishing minimax rates of convergence in noisy conditions and proposing an ERM method with deconvolution kernels to achieve these rates.

Contribution

It extends minimax rate results to discriminant analysis with errors in variables and proposes a practical ERM approach using deconvolution kernels.

Findings

Established minimax lower bounds for noisy discriminant analysis.

Proposed an ERM method based on deconvolution kernels to attain optimal rates.

Demonstrated the theoretical feasibility of fast convergence rates under measurement errors.

Abstract

The effect of measurement errors in discriminant analysis is investigated. Given observations $Z=X+\epsilon$, where $\epsilon$ denotes a random noise, the goal is to predict the density of $X$ among two possible candidates $f$ and $g$. We suppose that we have at our disposal two learning samples. The aim is to approach the best possible decision rule $G^\star$ defined as a minimizer of the Bayes risk. In the free-noise case $(\epsilon=0)$, minimax fast rates of convergence are well-known under the margin assumption in discriminant analysis (see \cite{mammen}) or in the more general classification framework (see \cite{tsybakov2004,AT}). In this paper we intend to establish similar results in the noisy case, i.e. when dealing with errors in variables. We prove minimax lower bounds for this problem and explain how can these rates be attained, using in particular an Empirical Risk Minimizer (ERM) method based on deconvolution kernel estimators.