Graph-based approximate message passing iterations

Abstract Approximate message passing (AMP) algorithms have become an important element of high-dimensional statistical inference, mostly due to their adaptability and concentration properties, the state evolution (SE) equations. This is demonstrated by growing number new iterations proposed for increasingly complex problems, ranging from multi-layer inference low-rank matrix estimation with elaborate priors. In this paper, we address following questions: there a structure underlying all AMP that unifies them in common framework? Can use such give modular proof equations, adaptable without reproducing each time full argument? We propose answer both questions, showing instances can be generically indexed oriented graph. enables unified interpretation these iterations, independent problem they solve, way composing arbitrarily. then show graph verify rigorous SE extending reach previous proofs proving recent heuristic derivations those Our naturally includes non-separable functions how existing refinements, as spatial coupling or matrix-valued variables, combined our framework.