Statistical learning with indirect observations - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year : 2012

Statistical learning with indirect observations

Abstract

Let $(X,Y)\in\mathcal{X}\times \mathcal{Y}$ be a random couple with unknown distribution $P$. Let $\GG$ be a class of measurable functions and $\ell$ a loss function. The problem of statistical learning deals with the estimation of the Bayes: $$g^*=\arg\min_{g\in\GG}\E_P \ell(g(X),Y). $$ In this paper, we study this problem when we deal with a contaminated sample $(Z_1,Y_1),\ldots , (Z_n,Y_n)$ of i.i.d. indirect observations. Each input $Z_i$, $i=1,\ldots ,n$ 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 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 \cite{kolt} for the direct case. It gives some insights into the effect of indirect measurements in the presence of fast rates of convergence.
Fichier principal
Vignette du fichier
noisystatlearn.pdf (366.03 Ko) Télécharger le fichier
Origin : Publisher files allowed on an open archive

Dates and versions

hal-00664125 , version 1 (29-01-2012)
hal-00664125 , version 2 (15-02-2012)
hal-00664125 , version 3 (10-07-2012)

Identifiers

Cite

Sébastien Loustau. Statistical learning with indirect observations. 2012. ⟨hal-00664125v3⟩
196 View
166 Download

Altmetric

Share

Gmail Facebook X LinkedIn More