Signal-Plus-Noise Decomposition of Nonlinear Spiked Random Matrix Models

TL;DR

This paper introduces a signal-plus-noise decomposition for nonlinear spiked random matrix models, revealing phase transitions and enabling analysis of signal recovery and community detection in transformed stochastic block models.

Contribution

It provides a novel decomposition technique for nonlinear models and applies it to new problems in signal recovery and community detection.

Findings

Identifies precise phase transitions in signal structure

Demonstrates applicability to signed signal recovery

Validates results with numerical simulations

Abstract

In this paper, we study a nonlinear spiked random matrix model where a nonlinear function is applied element-wise to a noise matrix perturbed by a rank-one signal. We establish a signal-plus-noise decomposition for this model and identify precise phase transitions in the structure of the signal components at critical thresholds of signal strength. To demonstrate the applicability of this decomposition, we then utilize it to study new phenomena in the problems of signed signal recovery in nonlinear models and community detection in transformed stochastic block models. Finally, we validate our results through a series of numerical simulations.