  Approximate message passing (AMP) is a low-cost iterative
parameter-estimation technique for certain high-dimensional linear systems with
non-Gaussian distributions. However, AMP only applies to independent
identically distributed (IID) transform matrices, but may become unreliable for
other matrix ensembles, especially for ill-conditioned ones. To handle this
difficulty, orthogonal/vector AMP (OAMP/VAMP) was proposed for general
right-unitarily-invariant matrices. However, the Bayes-optimal OAMP/VAMP
requires high-complexity linear minimum mean square error estimator. To solve
the disadvantages of AMP and OAMP/VAMP, this paper proposes a memory AMP
(MAMP), in which a long-memory matched filter is proposed for interference
suppression. The complexity of MAMP is comparable to AMP. The asymptotic
Gaussianity of estimation errors in MAMP is guaranteed by the orthogonality
principle. A state evolution is derived to asymptotically characterize the
performance of MAMP. Based on the state evolution, the relaxation parameters
and damping vector in MAMP are optimized. For all right-unitarily-invariant
matrices, the optimized MAMP converges to OAMP/VAMP, and thus is Bayes-optimal
if it has a unique fixed point. Finally, simulations are provided to verify the
validity and accuracy of the theoretical results.
