Random Linear Estimation With Rotationally-Invariant Designs: Asymptotics at High Temperature

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
$A^{\top} A$
and encompasses regimes of sufficiently low signal-to-noise ratio. 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.