### Abstract

Generalized Linear Models (GLMs), where a random vector $\mathbf{x}$ is observed through a noisy, possibly nonlinear, function of a linear transform $\mathbf{z}=\mathbf{Ax}$ arise in a range of applications in nonlinear filtering and regression. Approximate Message Passing (AMP) methods, based on loopy belief propagation, are a promising class of approaches for approximate inference in these models. AMP methods are computationally simple, general, and admit precise analyses with testable conditions for optimality for large i.i.d. transforms $\mathbf{A}$. However, the algorithms can easily diverge for general $\mathbf{A}$. This paper presents a convergent approach to the generalized AMP (GAMP) algorithm based on direct minimization of a large-system limit approximation of the Bethe Free Energy (LSL-BFE). The proposed method uses a double-loop procedure, where the outer loop successively linearizes the LSL-BFE and the inner loop minimizes the linearized LSL-BFE using the Alternating Direction Method of Multipliers (ADMM). The proposed method, called ADMM-GAMP, is similar in structure to the original GAMP method, but with an additional least-squares minimization. It is shown that for strictly convex, smooth penalties, ADMM-GAMP is guaranteed to converge to a local minima of the LSL-BFE, thus providing a convergent alternative to GAMP that is stable under arbitrary transforms. Simulations are also presented that demonstrate the robustness of the method for non-convex penalties as well.

Original language | Undefined |
---|---|

Article number | 1501.01797 |

Journal | arXiv |

State | Published - Jan 8 2015 |

### Keywords

- cs.IT
- math.IT

## Cite this

Rangan, S., Fletcher, A. K., Schniter, P., & Kamilov, U. (2015). Inference for generalized linear models via alternating directions and Bethe free energy minimization.

*arXiv*, [1501.01797]. http://arxiv.org/abs/1501.01797