  We propose an approach to multivariate nonparametric regression that
generalizes reduced rank regression for linear models. An additive model is
estimated for each dimension of a $q$-dimensional response, with a shared
$p$-dimensional predictor variable. To control the complexity of the model, we
employ a functional form of the Ky-Fan or nuclear norm, resulting in a set of
function estimates that have low rank. Backfitting algorithms are derived and
justified using a nonparametric form of the nuclear norm subdifferential.
Oracle inequalities on excess risk are derived that exhibit the scaling
behavior of the procedure in the high dimensional setting. The methods are
illustrated on gene expression data.
