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

Abstract

We study estimation in the linear model $y=A{\beta }^{\star }+\epsilon$, in a Bayesian setting where ${\beta }^{\star }$ has an entrywise i.i.d. prior and the design $A$ 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 of ${\beta }^{\star }$. In this work, we prove these conjectures for a general class of signal priors and for arbitrary rotationally-invariant designs $A$, 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.