  This paper presents a succinct derivation of the training and generalization
performance of a variety of high-dimensional ridge regression models using the
basic tools of random matrix theory and free probability. We provide an
introduction and review of recent results on these topics, aimed at readers
with backgrounds in physics and deep learning. Analytic formulas for the
training and generalization errors are obtained in a few lines of algebra
directly from the properties of the $S$-transform of free probability. This
allows for a straightforward identification of the sources of power-law scaling
in model performance. We compute the generalization error of a broad class of
random feature models. We find that in all models, the $S$-transform
corresponds to the train-test generalization gap, and yields an analogue of the
generalized-cross-validation estimator. Using these techniques, we derive
fine-grained bias-variance decompositions for a very general class of random
feature models with structured covariates. These novel results allow us to
discover a scaling regime for random feature models where the variance due to
the features limits performance in the overparameterized setting. We also
demonstrate how anisotropic weight structure in random feature models can limit
performance and lead to nontrivial exponents for finite-width corrections in
the overparameterized setting. Our results extend and provide a unifying
perspective on earlier models of neural scaling laws.
