Adaptive variable selection in nonparametric sparse regression

We study the problem of exact recovery of an unknown multivariate function $f$ observed in the continuous regression model. It is assumed that, in addition to some smoothness constraints, $f$ possesses an additive sparse structure determined by the sparsity index $\beta\in (0,1)$. As a consequence of the additive sparsity assumption, the recovery problem transforms to a variable selection problem. Conditions for exact variable selection are provided, and a family of asymptotically minimax variable selection procedures is constructed. The procedures are adaptive in the sparsity index $\beta$.