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

Matrix Dyson equation for correlated linearizations and test error of random features regression