Low-rank Matrix Estimation with Inhomogeneous Noise

TL;DR

This paper develops a rigorous theoretical framework for low-rank matrix estimation in inhomogeneous noise settings, extending existing models to more complex network structures like degree-corrected stochastic block models.

Contribution

It generalizes the spiked matrix model to inhomogeneous noise and derives a formula for the free energy, enabling analysis of signal detection thresholds in complex models.

Findings

Derived a rigorous free energy expression for inhomogeneous noise models

Extended analysis techniques from spin glasses to matrix estimation problems

Provided insights into signal detection in degree-corrected stochastic block models

Abstract

We study low-rank matrix estimation for a generic inhomogeneous output channel through which the matrix is observed. This generalizes the commonly considered spiked matrix model with homogeneous noise to include for instance the dense degree-corrected stochastic block model. We adapt techniques used to study multispecies spin glasses to derive and rigorously prove an expression for the free energy of the problem in the large size limit, providing a framework to study the signal detection thresholds. We discuss an application of this framework to the degree corrected stochastic block models.