Generalized Memory Approximate Message Passing

Generalized approximate message passing (GAMP) is a promising technique for unknown signal reconstruction of generalized linear models (GLM). However, it requires that the transformation matrix has independent and identically distributed (IID) entries. In this context, generalized vector AMP (GVAMP) is proposed for general unitarily-invariant transformation matrices but it has a high-complexity matrix inverse. To this end, we propose a universal generalized memory AMP (GMAMP) framework including the existing orthogonal AMP/VAMP, GVAMP, and MAMP as special instances. Due to the characteristics that local processors are all memory, GMAMP requires stricter orthogonality to guarantee the asymptotic IID Gaussianity and state evolution. To satisfy such orthogonality, local orthogonal memory estimators are established. The GMAMP framework provides a new principle toward building new advanced AMP-type algorithms. As an example, we construct a Bayes-optimal GMAMP (BO-GMAMP), which uses a low-complexity memory linear estimator to suppress the linear interference, and thus its complexity is comparable to GAMP. Furthermore, we prove that for unitarily-invariant transformation matrices, BO-GMAMP achieves the replica minimum (i.e., Bayes-optimal) MSE if it has a unique fixed point.