  Approximate message passing (AMP) is a low-cost iterative
parameter-estimation technique for certain high-dimensional linear systems with
non-Gaussian distributions. AMP only applies to independent identically
distributed (IID) transform matrices, but may become unreliable (e.g., perform
poorly or even diverge) for other matrix ensembles, especially for
ill-conditioned ones. To solve this issue, orthogonal/vector AMP (OAMP/VAMP)
was proposed for general right-unitarily-invariant matrices. However, the
Bayes-optimal OAMP/VAMP (BO-OAMP/VAMP) requires a high-complexity linear
minimum mean square error (MMSE) estimator. This prevents OAMP/VAMP from being
used in large-scale systems.
  To address the drawbacks of AMP and BO-OAMP/VAMP, this paper offers a memory
AMP (MAMP) framework based on the orthogonality principle, which ensures that
estimation errors in MAMP are asymptotically IID Gaussian. To realize the
required orthogonality for MAMP, we provide an orthogonalization procedure for
the local memory estimators. In addition, we propose a Bayes-optimal MAMP
(BO-MAMP), in which a long-memory matched filter is used for interference
suppression. The complexity of BO-MAMP is comparable to AMP. To asymptotically
characterize the performance of BO-MAMP, a state evolution is derived. The
relaxation parameters and damping vector in BO-MAMP are optimized based on
state evolution. Most crucially, the state evolution of the optimized BO-MAMP
converges to the same fixed point as that of the high-complexity BO-OAMP/VAMP
for all right-unitarily-invariant matrices, and achieves the Bayes optimal MSE
predicted by the replica method if its state evolution has a unique fixed
point. Finally, simulations are provided to verify the theoretical results'
validity and accuracy.
