Optimal Algorithms for the Inhomogeneous Spiked Wigner Model

TL;DR

This paper investigates the inhomogeneous spiked Wigner model, developing an AMP algorithm that matches the optimal Bayes performance and identifying a statistical-to-computational gap, along with a spectral method for phase transition detection.

Contribution

It introduces an AMP algorithm tailored for inhomogeneous noise profiles and establishes its optimality, also revealing a computational gap and proposing a spectral method for phase transition detection.

Findings

AMP algorithm matches Bayes optimal fixed-point equations

Existence of a statistical-to-computational gap

Spectral method accurately detects phase transition

Abstract

In this paper, we study a spiked Wigner problem with an inhomogeneous noise profile. Our aim in this problem is to recover the signal passed through an inhomogeneous low-rank matrix channel. While the information-theoretic performances are well-known, we focus on the algorithmic problem. We derive an approximate message-passing algorithm (AMP) for the inhomogeneous problem and show that its rigorous state evolution coincides with the information-theoretic optimal Bayes fixed-point equations. We identify in particular the existence of a statistical-to-computational gap where known algorithms require a signal-to-noise ratio bigger than the information-theoretic threshold to perform better than random. Finally, from the adapted AMP iteration we deduce a simple and efficient spectral method that can be used to recover the transition for matrices with general variance profiles. This spectral method matches the conjectured optimal computational phase transition.