Approximate Message Passing with Spectral Initialization for Generalized Linear Models

TL;DR

This paper introduces a spectral initialization method for approximate message passing algorithms in generalized linear models, providing a rigorous performance analysis in high dimensions and demonstrating improved practical applicability.

Contribution

It proposes a spectral initialization for AMP in generalized linear models and offers a rigorous high-dimensional performance analysis of this approach.

Findings

Spectral initialization enables AMP without prior ground-truth correlation.

The performance of AMP with spectral initialization is characterized in high dimensions.

Numerical results confirm the theoretical predictions and practical effectiveness.

Abstract

We consider the problem of estimating a signal from measurements obtained via a generalized linear model. We focus on estimators based on approximate message passing (AMP), a family of iterative algorithms with many appealing features: the performance of AMP in the high-dimensional limit can be succinctly characterized under suitable model assumptions; AMP can also be tailored to the empirical distribution of the signal entries, and for a wide class of estimation problems, AMP is conjectured to be optimal among all polynomial-time algorithms. However, a major issue of AMP is that in many models (such as phase retrieval), it requires an initialization correlated with the ground-truth signal and independent from the measurement matrix. Assuming that such an initialization is available is typically not realistic. In this paper, we solve this problem by proposing an AMP algorithm initialized with a spectral estimator. With such an initialization, the standard AMP analysis fails since the spectral estimator depends in a complicated way on the design matrix. Our main contribution is a rigorous characterization of the performance of AMP with spectral initialization in the high-dimensional limit. The key technical idea is to define and analyze a two-phase artificial AMP algorithm that first produces the spectral estimator, and then closely approximates the iterates of the true AMP. We also provide numerical results that demonstrate the validity of the proposed approach.