On Capacity Optimality of OAMP: Beyond IID Sensing Matrices and Gaussian Signaling

This paper investigates a large unitarily invariant system (LUIS) involving a unitarily invariant sensing matrix, an arbitrarily fixed signal distribution, and forward error control (FEC) coding. A universal Gram-Schmidt orthogonalization is considered for the construction of orthogonal approximate message passing (OAMP), which renders the results applicable to general prototypes without the differentiability restriction. For OAMP with Lipschitz continuous local estimators, we develop two variational single-input-single-output transfer functions, based on which we analyze the achievable rate of OAMP. Furthermore, when the state evolution of OAMP has a unique fixed point, we reveal that OAMP reaches the constrained capacity predicted by the replica method of the LUIS with an arbitrary signal distribution based on matched FEC coding. The replica method is rigorous for LUIS with Gaussian signaling and for certain sub-classes of LUIS with arbitrary signal distributions. Several area properties are established based on the variational transfer functions of OAMP. Meanwhile, we elaborate a replica constrained capacity-achieving coding principle for LUIS, based on which irregular low-density parity-check (LDPC) codes are optimized for binary signaling in the simulation results. We show that OAMP with the optimized codes has significant performance improvement over the un-optimized ones and the well-known Turbo linear MMSE algorithm. For quadrature phase-shift keying (QPSK) modulation, replica constrained capacity-approaching bit error rate (BER) performances are observed under various channel conditions.