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This is Part II of a two-part work on the estimation for a multi-layergeneralized linear model (ML-GLM) in large system limits. In Part I, we hadanalyzed the asymptotic performance of an exact MMSE estimator, and obtained aset of coupled equations that could characterize its MSE performance. To workaround the implementation difficulty of the exact estimator, this papercontinues to propose an approximate solution, ML-GAMP, which could be derivedby blending a moment-matching projection into the Gaussian approximated loopybelief propagation. The ML-GAMP estimator is then shown to enjoy a greatsimplicity in its implementation, where its per-iteration complexity is as lowas GAMP. Further analysis on its asymptotic performance also reveals that, inlarge system limits, its dynamical MSE behavior is fully characterized by a setof simple one-dimensional iterating equations, termed state evolution (SE).Interestingly, this SE of ML-GAMP share exactly the same fixed points with anexact MMSE estimator whose fixed points were obtained in Part I via a replicaanalysis. Given the Bayes-optimality of the exact implementation, this proposedestimator (if converged) is optimal in the MSE sense.

Estimation for High-Dimensional Multi-Layer Generalized Linear Model -- Part II: The ML-GAMP Estimator