Hypergraph partitioning is an important problem in machine learning, computer vision and network analytics. A widely used method for hypergraph partitioning relies on minimizing a normalized sum of the costs of partitioning hyperedges across clusters. Algorithmic solutions based on this approach assume that different partitions of a hyperedge incur the same cost. However, this assumption fails to leverage the fact that different subsets of vertices within the same hyperedge may have different structural importance. We hence propose a new hypergraph clustering technique, termed inhomogeneous hypergraph partitioning, which assigns different costs to different hyperedge cuts. We prove that inhomogeneous partitioning produces a quadratic approximation to the optimal solution if the inhomogeneous costs satisfy submodularity constraints. Moreover, we demonstrate that inhomogenous partitioning offers significant performance improvements in applications such as structure learning of rankings, subspace segmentation and motif clustering.

Neural Information Processing Systems http://nips.cc/

We introduce a new convex optimization problem, termed quadratic decomposable submodular function minimization. The problem is closely related to decomposable submodular function minimization and arises in many learning on graphs and hypergraphs settings, such as graph-based semi-supervised learning and PageRank. We approach the problem via a new dual strategy and describe an objective that may be optimized via random coordinate descent (RCD) methods and projections onto cones. We also establish the linear convergence rate of the RCD algorithm and develop efficient projection algorithms with provable performance guarantees. Numerical experiments in semi-supervised learning on hypergraphs confirm the efficiency of the proposed algorithm and demonstrate the significant improvements in prediction accuracy with respect to state-of-the-art methods.

Graph embedding has been intensively studied recently, due to the advance of various neural network models. Theoretical analyses and empirical studies have pushed forward the translation of discrete graph structures into distributed representation vectors, but seldom considered the reverse direction, i.e., generation of graphs from given related context spaces. Particularly, since graphs often become more meaningful when associated with semantic contexts (e.g., social networks of certain communities, gene networks of certain diseases), the ability to infer graph structures according to given semantic conditions could be of great value.

Graph embedding has been intensively studied recently, due to the advance of various neural network models. Theoretical analyses and empirical studies have pushed forward the translation of discrete graph structures into distributed representation vectors, but seldom considered the reverse direction, i.e., generation of graphs from given related context spaces. Particularly, since graphs often become more meaningful when associated with semantic contexts (e.g., social networks of certain communities, gene networks of certain diseases), the ability to infer graph structures according to given semantic conditions could be of great value.

Neural Information Processing Systems http://nips.cc/

We introduce a new approach to decomposable submodular function minimization (DSFM) that exploits incidence relations. Incidence relations describe which variables effectively influence the component functions, and when properly utilized, they allow for improving the convergence rates of DSFM solvers. Our main results include the precise parametrization of the DSFM problem based on incidence relations, the development of new scalable alternative projections and parallel coordinate descent methods and an accompanying rigorous analysis of their convergence rates.

Neural Information Processing Systems http://nips.cc/