Lower bounds on minimax rates for nonparametric regression with additive sparsity and smoothness

This paper uses information-theoretic techniques to determine minimax rates for estimating nonparametric sparse additive regression models under high-dimensional scaling. The first term reflects the difficulty of performing \emph{subset selection} and is independent of the Hilbert space \Hilb; the second term \LowerRateSq is an \emph{\s-dimensional estimation} term, depending only on the low dimension \s but not the ambient dimension \pdim, that captures the difficulty of estimating a sum of \s univariate functions in the Hilbert space \Hilb . The minimax rates are compared with rates achieved by an \ell_1 -penalty based approach, it can be shown that a certain \ell_1 -based approach achieves the minimax optimal rate.