Detection of sparse additive functions

Abstract

We study the problem of detection of a high-dimensional signal function in the white Gaussian noise model. As well as a smoothness assumption on the signal function, we assume an additive sparse condition on the latter. The detection problem is expressed in terms of a nonparametric hypothesis testing problem and it is solved according to the asymptotical minimax approach. The minimax test procedures are adaptive in the sparsity parameter for high sparsity case. We extend to the functional case the known results in the detection of sparse high-dimensional vectors. In particular, our asymptotic detection boundaries are derived from the same asymptotic relations as in the vector case.