Covariates-Adjusted Mixed-Membership Estimation: A Novel Network Model with Optimal Guarantees

TL;DR

This paper introduces a new covariates-adjusted mixed-membership network model that combines community structure and node similarities, achieving optimal estimation guarantees through innovative nonconvex optimization analysis.

Contribution

It proposes a novel model integrating covariates with mixed-membership estimation and develops a theoretical framework linking convex and nonconvex optimization for optimal accuracy guarantees.

Findings

Achieves optimal estimation accuracy for similarity matrix and mixed-membership.

Extends analysis to all convex loss functions beyond MLE and MSE.

Validates theoretical results with simulations and real data.

Abstract

This paper addresses the problem of mixed-membership estimation in networks, where the goal is to efficiently estimate the latent mixed-membership structure from the observed network. Recognizing the widespread availability and valuable information carried by node covariates, we propose a novel network model that incorporates both community information, as represented by the Degree-Corrected Mixed Membership (DCMM) model, and node covariate similarities to determine connections. We investigate the regularized maximum likelihood estimation (MLE) for this model and demonstrate that our approach achieves optimal estimation accuracy for both the similarity matrix and the mixed-membership, in terms of both the Frobenius norm and the entrywise loss. Since directly analyzing the original convex optimization problem is intractable, we employ nonconvex optimization to facilitate the analysis. A key contribution of our work is identifying a crucial assumption that bridges the gap between convex and nonconvex solutions, enabling the transfer of statistical guarantees from the nonconvex approach to its convex counterpart. Importantly, our analysis extends beyond the MLE loss and the mean squared error (MSE) used in matrix completion problems, generalizing to all the convex loss functions. Consequently, our analysis techniques extend to a broader set of applications, including ranking problems based on pairwise comparisons. Finally, simulation experiments validate our theoretical findings, and real-world data analyses confirm the practical relevance of our model.