Safe Element Screening for Submodular Function Minimization
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Submodular functions are discrete analogs of convex functions, which have applications in various fields, including machine learning, computer vision and signal processing. However, in large-scale applications, solving Submodular Function Minimization (SFM) problems remains challenging. In this paper, we make the first attempt to extend the emerging technique named screening in large-scale sparse learning to SFM for accelerating its optimization process. Specifically, we propose a novel safe element screening method---based on a careful studying of the relationships between SFM and the corresponding convex proximal problems, as well as the accurate estimation of the optimum of the proximal problem---to quickly identify the elements that are guaranteed to be included (we refer to them as active) or excluded (inactive) in the final optimal solution of SFM during the optimization process. By removing the inactive elements and fixing the active ones, the problem size can be dramatically reduced, leading to great savings in the computational cost without sacrificing accuracy. To the best of our knowledge, the proposed method is the first screening method in the fields of SFM and even combinatorial optimization, and thus points out a new direction for accelerating SFM algorithms. Experiment results on both synthetic and real datasets demonstrate the significant speedups gained by our screening method.
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