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Estimation and prediction problems for dense signals are often framed interms of minimax problems over highly symmetric parameter spaces. In thispaper, we study minimax problems over l2-balls for high-dimensional linearmodels with Gaussian predictors. We obtain sharp asymptotics for the minimaxrisk that are applicable in any asymptotic setting where the number ofpredictors diverges and prove that ridge regression is asymptotically minimax.Adaptive asymptotic minimax ridge estimators are also identified. Orthogonalinvariance is heavily exploited throughout the paper and, beyond serving as atechnical tool, provides additional insight into the problems considered here.Most of our results follow from an apparently novel analysis of an equivalentnon-Gaussian sequence model with orthogonally invariant errors. As with manydense estimation and prediction problems, the minimax risk studied here hasrate d/n, where d is the number of predictors and n is the number ofobservations; however, when d is roughly proportional to n the minimax risk isinfluenced by the spectral distribution of the predictors and is notablydifferent from the linear minimax risk for the Gaussian sequence model(Pinsker, 1980) that often appears in other dense estimation and predictionproblems.

Optimal Estimation and Prediction for Dense Signals in High-Dimensional Linear Models