Estimating Differential Latent Variable Graphical Models with Applications to Brain Connectivity
This work addresses brain connectivity analysis by improving differential network estimation, though it is incremental as it extends existing models with a new method.
The paper tackles the problem of estimating differential networks in latent variable Gaussian graphical models for two groups, proposing a two-stage nonconvex procedure that achieves minimax optimal statistical error and outperforms existing methods in experiments.
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.