Fixed points of generalized approximate message passing with arbitrary matrices

Abstract

Abstract-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.

High-Dimensional Macroeconomic Forecasting Using Message Passing Algorithms

“…This is a result of the Berry-Esseen central limit theorem which states that a sum of random variables converge to a Gaussian density; see a proof of this theorem inDonoho et al (2011). Given that the sumproduct equations involve products of random variables, rather than sums, derivations of GAMP based on this central limit theorem typically proceed by taking logarithms of equations (26)-(28). The marginal posterior p (β i |y) is then recovered by performing an exponential transformation of the log messages, and by normalizing so that the posterior integrates to one; see the Online Appendix, Section A, for details.…”

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“…The statistical behavior of AMP can be characterized theoretically for large iid subgaussian random feature matrices [15], and empirical results suggest that the theory holds more generally for certain types of nonrandom matrices [13]. One of the challenges with AMP, however, is that for arbitrary matrices convergence of the AMP iterations may require dampening [16] or serial updates [17]. Recent work has shown that stable points of the AMP iterations correspond to stationary points of an approximation to the Bethe free energy [16] and developed optimization methods which attempt to minimize the approximate Bethe free energy directly [18].…”

“…One of the challenges with AMP, however, is that for arbitrary matrices convergence of the AMP iterations may require dampening [16] or serial updates [17]. Recent work has shown that stable points of the AMP iterations correspond to stationary points of an approximation to the Bethe free energy [16] and developed optimization methods which attempt to minimize the approximate Bethe free energy directly [18]. While this leads to methods with guaranteed convergence, the statistical behavior of the solution is not fully understood for general matrices.…”

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