Orthogonal Approximate Message-Passing for Spatially Coupled Linear Models

Orthogonal approximate message-passing (OAMP) is proposed for signal recovery from right-orthogonally invariant linear measurements with spatial coupling. Conventional state evolution is generalized to a unified framework of state evolution for the spatial coupling and long-memory case. The unified framework is used to formulate the so-called Onsager correction in OAMP for spatially coupled systems. The state evolution recursion of Bayes-optimal OAMP is proved to converge for spatially coupled systems via Bayes-optimal long-memory OAMP and its state evolution. This paper proves the information-theoretic optimality of Bayes-optimal OAMP for noiseless spatially coupled systems with right-orthogonally invariant sensing matrices.