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Learning structured densities via infinite dimensional exponential families

Learning the structure of a probabilistic graphical models is a well studied problem in the machine learning community due to its importance in many applications. Current approaches are mainly focused on learning the structure under restrictive parametric assumptions, which limits the applicability of these methods. In this paper, we study the problem of estimating the structure of a probabilistic graphical model without assuming a particular parametric model. We consider probabilities that are members of an infinite dimensional exponential family, which is parametrized by a reproducing kernel Hilbert space (RKHS) H and its kernel $k$. One difficulty in learning nonparametric densities is evaluation of the normalizing constant. In order to avoid this issue, our procedure minimizes the penalized score matching objective. We show how to efficiently minimize the proposed objective using existing group lasso solvers. Furthermore, we prove that our procedure recovers the graph structure with high-probability under mild conditions. Simulation studies illustrate ability of our procedure to recover the true graph structure without the knowledge of the data generating process.

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