论文标题
关于主组件回归的预测错误的注释
A note on the prediction error of principal component regression in high dimensions
论文作者
论文摘要
我们分析了主成分回归（PCR）的预测误差，并证明了设计相应平方风险的高概率界限。我们的第一个主要结果表明，如果有效的等级条件持有有效的等级条件，则PCR的性能与通过其人口对应的经验主组件替换经验主成分获得的甲骨文方法相当。另一方面，如果违反了后一种情况，则经验特征值开始具有显着的向上偏见，从而导致PCR的自我诱导的正则化。我们的方法依赖于经验特征值，经验特征向量和高维度中主成分分析的过多风险。
We analyze the prediction error of principal component regression (PCR) and prove high probability bounds for the corresponding squared risk conditional on the design. Our first main result shows that PCR performs comparably to the oracle method obtained by replacing empirical principal components by their population counterparts, provided that an effective rank condition holds. On the other hand, if the latter condition is violated, then empirical eigenvalues start to have a significant upward bias, resulting in a self-induced regularization of PCR. Our approach relies on the behavior of empirical eigenvalues, empirical eigenvectors and the excess risk of principal component analysis in high-dimensional regimes.
