Fixed Points of Generalized Approximate Message Passing with Arbitrary Matrices
arXiv:1301.6295
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.
References in corpus (8)
- Bilinear Generalized Approximate Message Passing
- Adaptive Damping and Mean Removal for the Generalized Approximate Message Passing Algorithm
- Phase transitions and sample complexity in Bayes-optimal matrix factorization
- An Empirical-Bayes Approach to Recovering Linearly Constrained Non-Negative Sparse Signals
- Sparse Estimation with the Swept Approximated Message-Passing Algorithm
- Phase Diagram and Approximate Message Passing for Blind Calibration and Dictionary Learning
- Scalable Inference for Neuronal Connectivity from Calcium Imaging
- Approximate Message Passing with Consistent Parameter Estimation and Applications to Sparse Learning