Abstract

We present Vector-Space Markov Random Fields (VS-MRFs), a novel class of undirected graphical models where each variable can belong to an arbitrary vector space. VS-MRFs generalize a recent line of work on scalar-valued, uni-parameter exponential family and mixed graphical models, thereby greatly broadening the class of exponential families available (e.g., allowing multinomial and Dirichlet distributions). Specifically, VSMRFs are the joint graphical model distributions where the node-conditional distributions belong to generic exponential families with general vector space domains. We also present a sparsistent M -estimator for learning our class of MRFs that recovers the correct set of edges with high probability. We validate our approach via a set of synthetic data experiments as well as a realworld case study of over four million foods from the popular diet tracking app MyFitnessPal. Our results demonstrate that our algorithm performs well empirically and that VS-MRFs are capable of capturing and highlighting interesting structure in complex, real-world data. All code for our algorithm is open source and publicly available.