高维线性模型中密集信号的最优估计与预测
Optimal Estimation and Prediction for Dense Signals in High-Dimensional Linear Models
摘要：
密集信号的估计和预测问题常常被归结为高度对称参数空间上的极大极小问题。本文研究了具有高斯预测的高维线性模型在l2-球上的极大极小问题。我们得到尖锐的渐近极小极大风险，适用于任何渐近设置的预测变量的数量发散，并证明岭回归是渐近极小极大。自适应渐近极小极大岭估计也确定。正交不变性是大量利用整个文件，除了作为一个技术工具，提供了额外的洞察这里考虑的问题。我们的大部分结果来自一个明显新颖的分析，一个等效的非高斯序列模型与正交不变误差。与许多密集估计和预测问题一样，这里研究的极大极小风险率为d/n，其中d是预测因子的数量，n是观测值的数量;然而，当d与n大致成比例时，极大极小风险受预测因子的谱分布的影响，并且与高斯序列模型的线性极大极小风险显著不同（Pinsker，1980），经常出现在其他密集估计和预测问题。
英文摘要：
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.
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