Random linear estimation with rotationally-invariant designs: Asymptotics at high temperature
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We study estimation in the linear model y=Aβ^⋆+ϵ, in a Bayesian setting where β^⋆ 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 β^⋆. 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^⊤ 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.
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