Exact Recovery of Community Detection in dependent Gaussian Mixture Models

TL;DR

This paper investigates the conditions for exact community detection in Gaussian mixture models with complex, dependent noise structures, providing theoretical thresholds and optimization formulations.

Contribution

It introduces a constrained quadratic optimization approach for MLE with dependent Gaussian noise and establishes sharp recovery thresholds under various covariance assumptions.

Findings

Derived sufficient conditions for exact recovery with unknown and known community sizes.

Proved a sharp threshold for exact recovery in full-rank non-diagonal block models.

Identified a no-gap mechanism where sufficient and necessary conditions align asymptotically.

Abstract

We study exact recovery for community detection in a Gaussian mixture model with dependent and heterogeneous Gaussian noise. The noise covariance matrix $\Sigma$ may be non-diagonal and, in the general formulation, singular. In the singular case, we write the Gaussian likelihood on the support of the induced measure and show that the maximum likelihood estimator (MLE) is a constrained quadratic optimization problem involving the Moore--Penrose inverse. For general covariance structures, we obtain sufficient conditions for exact recovery of the MLE when the community sizes are unknown and when they are known. These conditions are driven by the $\Sigma$-whitened separation $L_\Sigma(x,y)$ together with local one-step comparison inequalities in the near-truth regime. Under the additional assumption that $\Sigma$ is invertible, we derive converse results showing failure of exact recovery when a large family of local perturbations has sufficiently nondegenerate Gaussian comparison statistics. We then analyze a full-rank non-diagonal block-covariance model, prove a sharp exact-recovery threshold in the unknown-size setting, and identify a general no-gap mechanism under which the sufficient and necessary conditions coincide asymptotically.