Statistical learning with indirect observations

Sébastien Loustau (1)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

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.

Mots clés: Statistical learning; Inverse problem; Classification; Deconvolution; Empirical processes; Fast rates

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Machine Learning [stat.ML]

Sébastien Loustau. Statistical learning with indirect observations. 2012. ⟨hal-00664125v3⟩