Universality laws for Gaussian mixtures in generalized linear models

A recent line of work in high-dimensional statistics working under the Gaussian mixture hypothesis has led to a number of results in the context of empirical risk minimization, Bayesian uncertainty quantification, separation of kernel methods and neural networks, ensembling and fluctuation of random features. Our main contribution is to characterize under which conditions the asymptotic joint statistics of this family depends (on a weak sense) only on the means and covariances of the class conditional features distribution P_{c} {\mathbf{x}} . This allows us to prove the universality of different quantities of interest, including training, generalization errors, as well as the geometrical properties and correlations of the estimators.