Estimating Differential Latent Variable Graphical Models with Applications to Brain Connectivity

TL;DR

This paper introduces a novel method for estimating differential Gaussian graphical models with latent variables, decomposing differences into sparse and low-rank parts, and demonstrates its effectiveness through theoretical guarantees and experiments.

Contribution

It presents a two-stage nonconvex optimization approach for differential latent variable graphical models, with proven convergence and optimal statistical error.

Findings

Method outperforms existing approaches in experiments.

Estimator converges linearly with minimax optimal error.

Effective in both synthetic and real data applications.

Abstract

Differential graphical models are designed to represent the difference between the conditional dependence structures of two groups, thus are of particular interest for scientific investigation. Motivated by modern applications, this manuscript considers an extended setting where each group is generated by a latent variable Gaussian graphical model. Due to the existence of latent factors, the differential network is decomposed into sparse and low-rank components, both of which are symmetric indefinite matrices. We estimate these two components simultaneously using a two-stage procedure: (i) an initialization stage, which computes a simple, consistent estimator, and (ii) a convergence stage, implemented using a projected alternating gradient descent algorithm applied to a nonconvex objective, initialized using the output of the first stage. We prove that given the initialization, the estimator converges linearly with a nontrivial, minimax optimal statistical error. Experiments on synthetic and real data illustrate that the proposed nonconvex procedure outperforms existing methods.