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2013 Inverse statistical learning
Sébastien Loustau
Electron. J. Statist. 7: 2065-2097 (2013). DOI: 10.1214/13-EJS838


Let $(X,Y)\in\mathcal{X}\times\mathcal{Y}$ be a random couple with unknown distribution $P$. Let $\mathcal{G}$ be a class of measurable functions and $\ell$ a loss function. The problem of statistical learning deals with the estimation of the Bayes: \[g^{*}=\arg\min_{g\in\mathcal{G}}\mathbb{E}_{P}\ell(g,(X,Y)).\] In this paper, we study this problem when we deal with a contaminated sample $(Z_{1},Y_{1}),\dots,(Z_{n},Y_{n})$ of i.i.d. indirect observations. Each input $Z_{i}$, $i=1,\dots,n$ is distributed from a density $Af$, where $A$ is a known compact linear operator and $f$ is the density of the direct input $X$.

We derive fast rates of convergence for the excess risk of empirical risk minimizers based on regularization methods, such as deconvolution kernel density estimators or spectral cut-off. These results are comparable to the existing fast rates in Koltchinskii (2006) for the direct case. It gives some insights into the effect of indirect measurements in the presence of fast rates of convergence.


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Sébastien Loustau. "Inverse statistical learning." Electron. J. Statist. 7 2065 - 2097, 2013.


Published: 2013
First available in Project Euclid: 20 August 2013

zbMATH: 1349.62102
MathSciNet: MR3091617
Digital Object Identifier: 10.1214/13-EJS838

Primary: 62G05
Secondary: 62H30

Keywords: ‎classification‎ , Deconvolution , fast rates , inverse problem , Statistical learning

Rights: Copyright © 2013 The Institute of Mathematical Statistics and the Bernoulli Society


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