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Differential graphical models are designed to represent the differencebetween the conditional dependence structures of two groups, thus are ofparticular interest for scientific investigation. Motivated by modernapplications, this manuscript considers an extended setting where each group isgenerated by a latent variable Gaussian graphical model. Due to the existenceof latent factors, the differential network is decomposed into sparse andlow-rank components, both of which are symmetric indefinite matrices. Weestimate 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 gradientdescent algorithm applied to a nonconvex objective, initialized using theoutput of the first stage. We prove that given the initialization, theestimator converges linearly with a nontrivial, minimax optimal statisticalerror. Experiments on synthetic and real data illustrate that the proposednonconvex procedure outperforms existing methods.

Estimating Differential Latent Variable Graphical Models with Applications to Brain Connectivity