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In this paper, we study the problem of inferring time-varying Markov randomfields (MRF), where the underlying graphical model is both sparse and changessparsely over time. Most of the existing methods for the inference oftime-varying MRFs rely on the regularized maximum likelihood estimation (MLE),that typically suffer from weak statistical guarantees and high computationaltime. Instead, we introduce a new class of constrained optimization problemsfor the inference of sparsely-changing MRFs. The proposed optimization problemis formulated based on the exact $\ell_0$ regularization, and can be solved innear-linear time and memory. Moreover, we show that the proposed estimatorenjoys a provably small estimation error. As a special case, we derive sharpstatistical guarantees for the inference of sparsely-changing Gaussian MRFs(GMRF) in the high-dimensional regime, showing that such problems can belearned with as few as one sample per time. Our proposed method is extremelyefficient in practice: it can accurately estimate sparsely-changing graphicalmodels with more than 500 million variables in less than one hour.

Scalable Inference of Sparsely-changing Markov Random Fields with Strong Statistical Guarantees