  We establish minimax optimal rates of convergence for estimation in a high
dimensional additive model assuming that it is approximately sparse. Our
results reveal an interesting phase transition behavior universal to this class
of high dimensional problems. In the {\it sparse regime} when the components
are sufficiently smooth or the dimensionality is sufficiently large, the
optimal rates are identical to those for high dimensional linear regression,
and therefore there is no additional cost to entertain a nonparametric model.
Otherwise, in the so-called {\it smooth regime}, the rates coincide with the
optimal rates for estimating a univariate function, and therefore they are
immune to the "curse of dimensionality".
