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In this work we propose a novel iterative estimation algorithm for linearobservation systems called S-AMP whose fixed points are the stationary pointsof the exact Gibbs free energy under a set of (first- and second-) momentconsistency constraints in the large system limit. S-AMP extends theapproximate message-passing (AMP) algorithm to general matrix ensembles. Thegeneralization is based on the S-transform (in free probability) of thespectrum of the measurement matrix. Furthermore, we show that the optimality ofS-AMP follows directly from its design rather than from solving a separateoptimization problem as done for AMP.

S-AMP: Approximate Message Passing for General Matrix Ensembles