TL;DR: Sparse graphical model structure estimators make restrictive assumptions. We show that empirical risk minimization can yield SOTA estimators for edge prediction across a wide range of graph structure distributions.
Abstract: We consider structure discovery of undirected graphical models from observational data. Inferring likely structures from few examples is a complex task often requiring the formulation of priors and sophisticated inference procedures. In the setting of Gaussian Graphical Models (GGMs) a popular estimator is a maximum likelihood objective with a penalization on the precision matrix. Adapting this estimator to capture domain-specific knowledge as priors or a new data likelihood requires great effort. In addition, structure recovery is an indirect consequence of the data-fit term. By contrast, it may be easier to generate training samples of data that arise from graphs with the desired structure properties. We propose here to leverage this latter source of information as training data to learn a function mapping from empirical covariance matrices to estimated graph structures. Learning this function brings two benefits: it implicitly models the desired structure or sparsity properties to form suitable priors, and it can be tailored to the specific problem of edge structure discovery, rather than maximizing data likelihood. We apply this framework to several real-world problems in structure discovery and show that it can be competitive to standard approaches such as graphical lasso, at a fraction of the execution speed. We use convolutional neural networks to parametrize our estimators due to the compositional structure of the problem. Experimentally, our learnable graph-discovery method trained on synthetic data generalizes well to different data: identifying relevant edges in real data, completely unknown at training time. We find that on genetics, brain imaging, and simulation data we obtain competitive(and generally superior) performance, compared with analytical methods.
Conflicts: inria.fr, ecp.fr, centralesupelec.fr, cs.toronto.edu, umontreal.ca, esat.kuleuven.be, google.com, ibm.com, ucl.ac.uk