  We consider the problem of variable selection in high-dimensional sparse
additive models. We focus on the case that the components belong to
nonparametric classes of functions. The proposed method is motivated by
geometric considerations in Hilbert spaces and consists of comparing the norms
of the projections of the data onto various additive subspaces. Under minimal
geometric assumptions, we prove concentration inequalities which lead to new
conditions under which consistent variable selection is possible. As an
application, we establish conditions under which a single component can be
estimated with the rate of convergence corresponding to the situation in which
the other components are known.
