We construct a classifier which attains the rate of convergence logn/n
under sparsity and margin assumptions. An approach close to the one met in
approximation theory for the estimation of function is used to obtain this
result. The idea is to develop the Bayes rule in a fundamental system of
L2([0,1]d) made of indicator of dyadic sets and to assume that
coefficients, equal to −1,0or1, belong to a kind of L1−ball. This
assumption can be seen as a sparsity assumption, in the sense that the
proportion of coefficients non equal to zero decreases as "frequency" grows.
Finally, rates of convergence are obtained by using an usual trade-off between
a bias term and a variance term