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Submodular functions have been studied extensively in machine learning anddata mining. In particular, the optimization of submodular functions over theinteger lattice (integer submodular functions) has recently attracted muchinterest, because this domain relates naturally to many practical problemsettings, such as multilabel graph cut, budget allocation and revenuemaximization with discrete assignments. In contrast, the use of these functionsfor probabilistic modeling has received surprisingly little attention so far.In this work, we firstly propose the Generalized Multilinear Extension, acontinuous DR-submodular extension for integer submodular functions. We studycentral properties of this extension and formulate a new probabilistic modelwhich is defined through integer submodular functions. Then, we introduce ablock-coordinate ascent algorithm to perform approximate inference for thoseclass of models. Finally, we demonstrate its effectiveness and viability onseveral real-world social connection graph datasets with integer submodularobjectives.

From Sets to Multisets: Provable Variational Inference for Probabilistic Integer Submodular Models