Adaptive variable selection in nonparametric sparse additive models

We consider the problem of recovery of an unknown multivariate signal f observed in a d-dimensional Gaussian white noise model of intensity ε. We assume that f belongs to a class of smooth functions in L₂([0, 1]^d) and has an additive sparse structure determined by the parameter s, the number of non-zero univariate components contributing to f. We are interested in the case when d = d_ε →∞as ε → 0 and the parameter s stays “small” relative to d. With these assumptions, the recovery problem in hand becomes that of determining which sparse additive components are non-zero. Attempting to reconstruct most, but not all, non-zero components of f, we arrive at the problem of almost full variable selection in high-dimensional regression. For two different choices of a class of smooth functions, we establish conditions under which almost full variable selection is possible, and provide a procedure that achieves this goal. Our procedure is the best possible (in the asymptotically minimax sense) for selecting most non-zero components of f. Moreover, it is adaptive in the parameter s. In addition to that, we complement the findings of [17] by obtaining an adaptive exact selector for the class of infinitely-smooth functions. Our theoretical results are illustrated with numerical experiments.