Fast and Scalable Learning of Sparse Changes in High-Dimensional Gaussian Graphical Model Structure

We focus on the problem of estimating the change in the dependency structures of two $p$-dimensional Gaussian Graphical models (GGMs). Previous studies for sparse change estimation in GGMs involve expensive and difficult non-smooth optimization. We propose a novel method, DIFFEE for estimating DIFFerential networks via an Elementary Estimator under a high-dimensional situation. DIFFEE is solved through a faster and closed form solution that enables it to work in large-scale settings. We conduct a rigorous statistical analysis showing that surprisingly DIFFEE achieves the same asymptotic convergence rates as the state-of-the-art estimators that are much more difficult to compute. Our experimental results on multiple synthetic datasets and one real-world data about brain connectivity show strong performance improvements over baselines, as well as significant computational benefits.Comment: 20pages, 6 figures, 10 tables; at AISTAT 201

Learning the piece-wise constant graph structure of a varying Ising model

This work focuses on the estimation of multiple change-points in a time-varying Ising model that evolves piece-wise constantly. The aim is to identify both the moments at which significant changes occur in the Ising model, as well as the underlying graph structures. For this purpose, we propose to estimate the neighborhood of each node by maximizing a penalized version of its conditional log-likelihood. The objective of the penalization is twofold: it imposes sparsity in the learned graphs and, thanks to a fused-type penalty, it also enforces them to evolve piece-wise constantly. Using few assumptions, we provide two change-points consistency theorems. Those are the first in the context of unknown number of change-points detection in time-varying Ising model. Finally, experimental results on several synthetic datasets and a real-world dataset demonstrate the performance of our method.Comment: 18 pages (9 pages for Appendix), 4 figures, 2 table

Differential Network Learning Beyond Data Samples

Learning the change of statistical dependencies between random variables is an essential task for many real-life applications, mostly in the high dimensional low sample regime. In this paper, we propose a novel differential parameter estimator that, in comparison to current methods, simultaneously allows (a) the flexible integration of multiple sources of information (data samples, variable groupings, extra pairwise evidence, etc.), (b) being scalable to a large number of variables, and (c) achieving a sharp asymptotic convergence rate. Our experiments, on more than 100 simulated and two real-world datasets, validate the flexibility of our approach and highlight the benefits of integrating spatial and anatomic information for brain connectome change discovery and epigenetic network identification.Comment: 9 pages of main draft; 25 pages of Appendix; 5 Tables ; 14 Figures ; Learning of Structure Difference between Two Graphical Model