Fixed points of generalized approximate message passing with arbitrary matrices

Sundeep Rangan, Philip Schniter, Erwin Riegler, Alyson K. Fletcher, Volkan Cevher

Research output: Contribution to journalArticlepeer-review


The estimation of a random vector with independent components passed through a linear transform followed by a componentwise (possibly nonlinear) output map arises in a range of applications. Approximate message passing (AMP) methods, based on Gaussian approximations of loopy belief propagation, have recently attracted considerable attention for such problems. For large random transforms, these methods exhibit fast convergence and admit precise analytic characterizations with testable conditions for optimality, even for certain non-convex problem instances. However, the behavior of AMP under general transforms is not fully understood. In this paper, we consider the generalized AMP (GAMP) algorithm and relate the method to more common optimization techniques. This analysis enables a precise characterization of the GAMP algorithm fixed points that applies to arbitrary transforms. In particular, we show that the fixed points of the so-called max-sum GAMP algorithm for MAP estimation are critical points of a constrained maximization of the posterior density. The fixed points of the sum-product GAMP algorithm for estimation of the posterior marginals can be interpreted as critical points of a certain free energy.

Original languageEnglish (US)
Article number7600404
Pages (from-to)7464-7474
Number of pages11
JournalIEEE Transactions on Information Theory
Issue number12
StatePublished - Dec 2016


  • ADMM
  • Message passing
  • belief propagation
  • compressed sensing
  • variational optimization

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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