Gaussian-Process Dynamics of Diagonal Expectation Propagation under Variance-Profile Gaussian Measurements

Abstract: State-evolution analyses of approximate-message-passing and expectation-propagation-type algorithms rely on an effective-channel principle: after a suitable Onsager, orthogonal, or extrinsic correction, the nonlinear module receives a fresh scalar Gaussian observation. This paper studies this principle for diagonal expectation propagation under variance-profile Gaussian sensing matrices. The model preserves Gaussian conditioning, but removes the isotropy that supports the usual scalar decoupling arguments. We prove a finite-time large-system description in which the linear EP module remains Gaussian at the coordinate level, but is generally not a fresh scalar channel. Instead, the residuals form a coordinate-dependent Gaussian process whose covariance is shaped by the variance profile and by the finite linear history of the algorithm. The standard diagonal EP cavity cancels the instantaneous response of the incoming message, but may leave a component predictable from past residuals. We characterize this process through a conditioned matrix-Dyson-equation deterministic equivalent and a Schur-complement representation of the linear module. A Gaussian-regression decomposition then separates the predictable memory from the orthogonal innovation and yields an oracle state-evolution-level correction. Thus, under variance-profile measurements, the limiting object for diagonal EP is a Gaussian-process dynamics with profile-dependent memory rather than the conventional fresh-noise scalar state evolution.