  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 memory AMP (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 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.
