Gaussian Mixture Models for Stochastic Block Models with Non-Vanishing Noise

Abstract

Community detection tasks have received a lot of attention across statistics, machine learning, and information theory with a large body of work concentrating on theoretical guarantees for the stochastic block model. One line of recent work has focused on modeling the spectral embedding of a network using Gaussian mixture models (GMMs) in scaling regimes where the ability to detect community memberships improves with the size of the network. However, these regimes are not very realistic. This paper provides tractable methodology motivated by new theoretical results for networks with non-vanishing noise. We present a procedure for community detection using GMMs that incorporates certain truncation and shrinkage effects that arise in the non-vanishing noise regime. We provide empirical validation of this new representation using both simulated and real-world data.

Mutual Information in Community Detection with Covariate Information and Correlated Networks

“…In this setting, we found that the BP has convergence issues and so we compare our theoretical results with the empirical performance of a spectral method [33] applied to a linear combination of the adjacency matrices. Specifically, we obtain estimates of the community labels using the following procedure.…”

“…Next, we retain the eigenvectors associated with the second and third leading eigenvalues in the spectral decomposition ofG. The relationship between these eigenvectors and the node labels is characterized using a Gaussian mixture model (GMM) approach described in [33], evaluated withR. Figure 2 shows the MSE as a function of the SBM parameter r. The solid blue line corresponds to the trace of the heuristic upper bound to the MMSE for two correlated networks computed from Theorem 2, and the red line corresponds to the upper bound for a single network.…”

“…Note that the conditional expectation of G given X is comparable to that of a single network with R = √ 2rI. Next, we retain the eigenvectors associated with the second and third leading eigenvalues in the spectral decomposition of G. The relationship between these eigenvectors and the node labels is characterized using a Gaussian mixture model (GMM) approach described in [33], evaluated with R.…”

Latent Community Adaptive Network Regression

“…Spectral methods leverage an embedding of the adjacency matrix in a Euclidean space and then estimate communities by clustering in the embedded space. There are many variations on spectral methods (Rohe et al, 2011;Binkiewicz et al, 2017b;Abbe, 2018;Mathews et al, 2019;Suwan et al, 2016), but here we concentrate on a normalized graph Laplacian embedding. Specifically, let ).…”

GMM discriminant analysis with noisy label for each class

“…Noisy label problem have been investigated for a long time in the machine learning literature and label noise-robust algorithms have numerous applications in medical image processing, spam filtering [1,2,3], Alzheimer disease prediction [1], gene expression classification [4] , text processing [5,6,7,8,9,10,11], image recognition [12,13,14,15,16,17]. Noisy labels are introduced by expert error and other unknown and unexpected factors.…”