Random linear estimation with rotationally-invariant designs: Asymptotics at high temperature

Yufan Li, Zhou Fan, Subhabrata Sen, Yihong Wu

Abstract: We study estimation in the linear model , in a Bayesian setting where the signal has an entrywise i.i.d. prior and the design matrix $A$ is rotationally-invariant in law. In the large system limit as dimension and sample size increase proportionally, a set of related conjectures have been postulated for the asymptotic mutual information, Bayes-optimal mean squared error, and TAP mean-field equations that characterize the Bayes posterior mean. In this work, we prove these conjectures for a general class of signal priors and for arbitrary rotationally-invariant designs, under a "high-temperature" condition that restricts the range of eigenvalues of $A^\top A$. Our proof uses a conditional second-moment method argument, where we condition on the iterates of a version of the Vector AMP algorithm for solving the TAP mean-field equations.