Classification with minimax fast rates for classes of Bayes rules with sparse representation

Abstract

We consider the classification problem on the cube [0,1]^d when the Bayes rule is known to belong to some new functions classes. These classes are made of prediction rules satisfying some conditions regarding their coefficients when developed over the (overcomplete) basis of indicator functions of dyadic cubes of [0,1]^d. The main concern of the paper is on the thorough analysis of the approximation term, which is in general bypassed in the classification literature. An adaptive classifier is designed to achieve the minimax rate of convergence (up to a logarithmic factor) over these functions classes. Lower bounds on the convergence rate over these classes are established when the underlying marginal of the design is comparable to the Lebesgue measure. Connections with some existing models for classification (RKHS and “boundary fragements”) are established.