Anisotropic oracle inequalities in noisy quantization

Abstract

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 $Z_i=X_i+{\epsilon }_i,i=1,\dots ,n$, where $X_i$ are i.i.d. with density $f$ and ${\epsilon }_i$ are i.i.d. with density $\eta$. These rates depend on the geometry of the density $f$ and the asymptotic behaviour of the characteristic function of $\eta$. 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.

30 pages. arXiv admin note: text overlap with arXiv:1205.1417

References in corpus (2)

• Statistical performance of support vector machines
• Minimax fast rates for discriminant analysis with errors in variables

Cited by in corpus (2)

• The algorithm of noisy k-means
• Noisy classification with boundary assumptions