Vector Approximate Message Passing for the Generalized Linear Model

TL;DR

This paper extends the Vector Approximate Message Passing (VAMP) algorithm to generalized linear models, providing a more robust and efficient inference method for large, ill-conditioned matrices in various applications.

Contribution

The paper introduces GLM-VAMP, an extension of VAMP for generalized linear models, improving robustness over existing GAMP methods for ill-conditioned matrices.

Findings

GLM-VAMP outperforms damped GAMP in robustness to ill-conditioning.

Numerical experiments confirm the effectiveness of GLM-VAMP.

The method applies to a wide range of applications involving nonlinear observations.

Abstract

The generalized linear model (GLM), where a random vector $\boldsymbol{x}$ is observed through a noisy, possibly nonlinear, function of a linear transform output $\boldsymbol{z}=\boldsymbol{Ax}$, arises in a range of applications such as robust regression, binary classification, quantized compressed sensing, phase retrieval, photon-limited imaging, and inference from neural spike trains. When $\boldsymbol{A}$ is large and i.i.d. Gaussian, the generalized approximate message passing (GAMP) algorithm is an efficient means of MAP or marginal inference, and its performance can be rigorously characterized by a scalar state evolution. For general $\boldsymbol{A}$, though, GAMP can misbehave. Damping and sequential-updating help to robustify GAMP, but their effects are limited. Recently, a "vector AMP" (VAMP) algorithm was proposed for additive white Gaussian noise channels. VAMP extends AMP's guarantees from i.i.d. Gaussian $\boldsymbol{A}$ to the larger class of rotationally invariant $\boldsymbol{A}$. In this paper, we show how VAMP can be extended to the GLM. Numerical experiments show that the proposed GLM-VAMP is much more robust to ill-conditioning in $\boldsymbol{A}$ than damped GAMP.