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We introduce a new convex optimization problem, termed quadratic decomposablesubmodular function minimization (QDSFM), which allows to model a number oflearning tasks on graphs and hypergraphs. The problem exhibits close ties todecomposable submodular function minimization (DSFM), yet is much morechallenging to solve. We approach the problem via a new dual strategy andformulate an objective that can be optimized through a number of double-loopalgorithms. The outer-loop uses either random coordinate descent (RCD) oralternative projection (AP) methods, for both of which we prove linearconvergence rates. The inner-loop computes projections onto cones generated bybase polytopes of the submodular functions, via the modified min-norm-point orFrank-Wolfe algorithm. We also describe two new applications of QDSFM:hypergraph-adapted PageRank and semi-supervised learning. The proposedhypergraph-based PageRank algorithm can be used for local hypergraphpartitioning, and comes with provable performance guarantees. Forhypergraph-adapted semi-supervised learning, we provide numerical experimentsdemonstrating the efficiency of our QDSFM solvers and their significantimprovements on prediction accuracy when compared to state-of-the-art methods.

Quadratic Decomposable Submodular Function Minimization: Theory and Practice (Computation and Analysis of PageRank over Hypergraphs)