Memory AMP

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 (e.g. perform poorly or even diverge) for other matrix ensembles, especially for ill-conditioned ones. Orthogonal/vector AMP (OAMP/VAMP) was proposed for general right-unitarily-invariant matrices to handle this difficulty. However, the Bayes-optimal OAMP/VAMP requires a high-complexity linear minimum mean square error (MMSE) estimator. This limits the application of OAMP/VAMP to large-scale systems. To solve the disadvantages of AMP and OAMP/VAMP, this paper proposes a memory AMP (MAMP) framework under an orthogonality principle, which guarantees the asymptotic IID Gaussianity of estimation errors in MAMP. We present an orthogonalization procedure for the local memory estimators to realize the required orthogonality for MAMP. Furthermore, we propose a Bayes-optimal MAMP (BO-MAMP), in which a long-memory matched filter is proposed for interference suppression. The complexity of BO-MAMP is comparable to AMP. A state evolution is derived to asymptotically characterize the performance of BO-MAMP. Based on state evolution, the relaxation parameters and damping vector in BO-MAMP are optimized. For all right-unitarily-invariant matrices, the optimized BO-MAMP converges to the high-complexity 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.