Anisotropic oracle inequalities in noisy quantization

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The effect of errors in variables in quantization is investigated. We prove general exact and non-exact oracle inequalities with fast rates for an empirical minimization based on a noisy sample Zᵢ=Xᵢ+εᵢ,i=1,,n, where Xᵢ are i.i.d. with density f and εᵢ are i.i.d. with density η. These rates depend on the geometry of the density f and the asymptotic behaviour of the characteristic function of η. This general study can be applied to the problem of k-means clustering with noisy data. For this purpose, we introduce a deconvolution k-means stochastic minimization which reaches fast rates of convergence under standard Pollard's regularity assumptions.