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Approximate message passing (AMP) has emerged both as a popular class ofiterative algorithms and as a powerful analytic tool in a wide range ofstatistical estimation problems and statistical physics models. A wellestablished line of AMP theory proves Gaussian approximations for the empiricaldistributions of the AMP iterate in the high dimensional limit, under the GOErandom matrix model and its variants.  This paper provides a non-asymptotic, leave-one-out representation for theAMP iterate that holds under a broad class of Gaussian random matrix modelswith general variance profiles. In contrast to the typical AMP theory thatdescribes the empirical distributions of the AMP iterate via a low dimensionalstate evolution, our leave-one-out representation yields an intrinsically highdimensional state evolution formula which provides non-asymptoticcharacterizations for the possibly heterogeneous, entrywise behavior of the AMPiterate under the prescribed random matrix models.  To exemplify some distinct features of our AMP theory in applications, weanalyze, in the context of regularized linear estimation, the precisestochastic behavior of the Ridge estimator for independent and non-identicallydistributed observations whose covariates exhibit general variance profiles. Wefind that its finite-sample distribution is characterized via a weighted Ridgeestimator in a heterogeneous Gaussian sequence model. Notably, in contrast tothe i.i.d. sampling scenario, the effective noise and regularization are nowfull dimensional vectors determined via a high dimensional system of equations.  Our leave-one-out method of proof differs significantly from the widelyadopted conditioning approach for rotational invariant ensembles, and reliesinstead on an inductive method that utilizes almost solely integration-by-partsand concentration techniques.

A leave-one-out approach to approximate message passing