We consider linear regression problems with a varying number of random projections, where we provably exhibit a double descent curve for a fixed prediction problem, with a high-dimensional analysis based on random matrix theory. We first consider the ridge regression estimator and review earlier results using classical notions from non-parametric statistics, namely degrees of freedom, also known as effective dimensionality. We then compute asymptotic equivalents of the generalization performance (in terms of squared bias and variance) of the minimum norm least-squares fit with random projections, providing simple expressions for the double descent phenomenon.

翻译：我们研究了具有可变数量随机投影的线性回归问题，基于随机矩阵理论的高维分析，我们严格证明了对固定预测问题存在的双重下降曲线。首先考虑岭回归估计量，并利用非参数统计中的经典概念（即自由度，亦称为有效维度）回顾了既有研究成果。随后计算了最小范数最小二乘拟合在随机投影下的泛化性能（以平方偏差和方差衡量）的渐近等价形式，为双重下降现象提供了简洁表达式。