High-dimensional linear regression has been thoroughly studied in the context of independent and identically distributed data. We propose to investigate high-dimensional regression models for independent but non-identically distributed data. To this end, we suppose that the set of observed predictors (or features) is a random matrix with a variance profile and with dimensions growing at a proportional rate. Assuming a random effect model, we study the predictive risk of the ridge estimator for linear regression with such a variance profile. In this setting, we provide deterministic equivalents of this risk and of the degree of freedom of the ridge estimator. For certain class of variance profile, our work highlights the emergence of the well-known double descent phenomenon in high-dimensional regression for the minimum norm least-squares estimator when the ridge regularization parameter goes to zero. We also exhibit variance profiles for which the shape of this predictive risk differs from double descent. The proofs of our results are based on tools from random matrix theory in the presence of a variance profile that have not been considered so far to study regression models. Numerical experiments are provided to show the accuracy of the aforementioned deterministic equivalents on the computation of the predictive risk of ridge regression. We also investigate the similarities and differences that exist with the standard setting of independent and identically distributed data.

翻译：高维线性回归已在独立同分布数据的背景下得到深入研究。我们提出研究独立但非同分布数据的高维回归模型。为此，我们假设观测预测变量（或特征）集合是一个具有方差分布的随机矩阵，其维度以成比例的速度增长。在随机效应模型的假设下，我们研究了具有此类方差分布的线性回归中岭估计量的预测风险。在此设定下，我们给出了该风险及岭估计量自由度的确定性等价量。对于特定类别的方差分布，我们的研究揭示了当岭正则化参数趋于零时，最小范数最小二乘估计量在高维回归中出现的著名双下降现象。我们还展示了某些方差分布会导致预测风险形态与双下降现象不同。我们结果的证明基于具有方差分布的随机矩阵理论工具，这些工具此前尚未被用于研究回归模型。数值实验证明了上述确定性等价量在计算岭回归预测风险时的准确性。我们还探讨了其与标准独立同分布数据设定之间的异同。