  Estimation and prediction problems for dense signals are often framed in
terms of minimax problems over highly symmetric parameter spaces. In this
paper, we study minimax problems over l2-balls for high-dimensional linear
models with Gaussian predictors. We obtain sharp asymptotics for the minimax
risk that are applicable in any asymptotic setting where the number of
predictors diverges and prove that ridge regression is asymptotically minimax.
Adaptive asymptotic minimax ridge estimators are also identified. Orthogonal
invariance is heavily exploited throughout the paper and, beyond serving as a
technical tool, provides additional insight into the problems considered here.
Most of our results follow from an apparently novel analysis of an equivalent
non-Gaussian sequence model with orthogonally invariant errors. As with many
dense estimation and prediction problems, the minimax risk studied here has
rate d/n, where d is the number of predictors and n is the number of
observations; however, when d is roughly proportional to n the minimax risk is
influenced by the spectral distribution of the predictors and is notably
different from the linear minimax risk for the Gaussian sequence model
(Pinsker, 1980) that often appears in other dense estimation and prediction
problems.
