标题： 岭回归中的风险与相关样本的交叉验证 显示英文标题

标题： Risk and cross validation in ridge regression with correlated samples

摘要： 近年来，我们在高维岭回归的理解方面取得了显著进展，但现有的理论假设训练样本是独立的。 通过利用随机矩阵理论和自由概率的技术，我们提供了当数据点具有任意相关性时，岭回归的内部和外部样本风险的精确渐近分析。 我们证明在这一设定下，广义交叉验证估计量（GCV）无法正确预测外部样本风险。 然而，在噪声残差与数据点具有相同相关性的条件下，可以对GCV进行修改，得到一个高效计算的无偏估计量，在高维极限下集中，我们将其称为CorrGCV。 我们进一步将渐近分析扩展到测试点与训练集具有非平凡相关性的场景，这种场景在时间序列预测中经常遇到。 假设已知时间序列的相关结构，这再次产生了一个GCV估计量的扩展，并精确描述了此类测试点在多大程度上会导致长期风险的过于乐观的预测。 我们在各种高维数据中验证了我们理论的预测。 显示英文摘要

摘要： Recent years have seen substantial advances in our understanding of high-dimensional ridge regression, but existing theories assume that training examples are independent. By leveraging techniques from random matrix theory and free probability, we provide sharp asymptotics for the in- and out-of-sample risks of ridge regression when the data points have arbitrary correlations. We demonstrate that in this setting, the generalized cross validation estimator (GCV) fails to correctly predict the out-of-sample risk. However, in the case where the noise residuals have the same correlations as the data points, one can modify the GCV to yield an efficiently-computable unbiased estimator that concentrates in the high-dimensional limit, which we dub CorrGCV. We further extend our asymptotic analysis to the case where the test point has nontrivial correlations with the training set, a setting often encountered in time series forecasting. Assuming knowledge of the correlation structure of the time series, this again yields an extension of the GCV estimator, and sharply characterizes the degree to which such test points yield an overly optimistic prediction of long-time risk. We validate the predictions of our theory across a variety of high dimensional data.