This paper develops some theory of the matrix Dyson equation (MDE) for correlated linearizations and uses it to solve a problem on asymptotic deterministic equivalent for the test error in random features regression. The theory developed for the correlated MDE includes existence-uniqueness, spectral support bounds, and stability properties of the MDE. This theory is new for constructing deterministic equivalents for pseudoresolvents of a class of correlated linear pencils. In the application, this theory is used to give a deterministic equivalent of the test error in random features ridge regression, in a proportional scaling regime, wherein we have conditioned on both training and test datasets.

翻译：本文发展了关联线性化情形下矩阵戴森方程（MDE）的部分理论，并将其应用于求解随机特征回归中测试误差的渐近确定性等价问题。所建立的关联MDE理论包括解的存在唯一性、谱支撑界及稳定性性质。该理论为构造一类关联线性束的伪预解式的确定性等价提供了新方法。在应用层面，本文利用该理论给出了比例缩放机制下（其中训练集与测试集均为条件样本）随机特征岭回归测试误差的确定性等价。