Vector-Space Markov Random Fields via Exponential Families
This work addresses the need for more flexible graphical models in machine learning, particularly for complex, real-world data, though it appears incremental as it builds on prior exponential family and mixed graphical models.
The authors tackled the problem of modeling variables in arbitrary vector spaces by introducing Vector-Space Markov Random Fields (VS-MRFs), which generalize existing graphical models to allow for broader exponential families like multinomial and Dirichlet distributions, and demonstrated empirical performance with a real-world case study on over four million foods from MyFitnessPal.
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, VS-MRFs 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 real-world 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.