Optimal Algorithms for the Inhomogeneous Spiked Wigner Model

Alexander Pak, Justin Ko, Florent Krzakala

Keywords: Spectral Method, Community detection, Wigner Spike model, Random Matrix, BBP transition, Approximate Message Passing, Spin glasses, Statistical Physics

TL;DR: To get a BBP transition in the spike model with Inhomogeneous noise, you should do PCA on *this* matrix, not the original one!

Abstract: 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. First, 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. Second, we deduce a simple and efficient spectral method that outperforms PCA and is shown to match the information-theoretic transition.

Submission Number: 13074