Classification by sparse additive models

We consider (nonparametric) sparse additive models (SpAM) for classification. The design of a SpAM classifier is based on minimizing the logistic loss with a sparse group Lasso and more general sparse group Slope-type penalties on the coefficients of univariate components' expansions in orthonormal series (e.g., Fourier or wavelets). The resulting classifiers are inherently adaptive to the unknown sparsity and smoothness. We show that under certain sparse group restricted eigenvalue condition the sparse group Lasso classifier is nearly-minimax (up to log-factors) within the entire range of analytic, Sobolev and Besov classes while the sparse group Slope classifier achieves the exact minimax order (without the extra log-factors) for sparse and moderately dense setups. The performance of the proposed classifier is illustrated on the real-data example.