Sparse additive regression on a regular lattice

We consider estimation in a sparse additive regression model with the design points on a regular lattice. We establish the minimax convergence rates over Sobolev classes and propose a Fourier-based rate optimal estimator which is adaptive to the unknown sparsity and smoothness of the response function. The estimator is derived within a Bayesian formalism but can be naturally viewed as a penalized maximum likelihood estimator with the complexity penalties on the number of non-zero univariate additive components of the response and on the numbers of the non-zero coefficients of their Fourer expansions. We compare it with several existing counterparts and perform a short simulation study to demonstrate its performance.