In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

Input: Consider Input: Normalized graph cut (NGC) has become a popular research topic due to its wide applications in a large variety of areas like machine learning and very large scale integration (VLSI) circuit design. Most of traditional NGC methods are based on pairwise relationships (similarities). However, in real-world applications relationships among the vertices (objects) may be more complex than pairwise, which are typically represented as hyperedges in hypergraphs. Thus, normalized hypergraph cut (NHC) has attracted more and more attention. Existing NHC methods cannot achieve satisfactory performance in real applications. In this paper, we propose a novel relaxation approach, which is called relaxed NHC (RNHC), to solve the NHC problem. Our model is defined as an optimization problem on the Stiefel manifold. To solve this problem, we resort to the Cayley transformation to devise a feasible learning algorithm. Experimental results on a set of large hypergraph benchmarks for clustering and partitioning in VLSI domain show that RNHC can outperform the state-of-the-art methods.

Output: A New Relaxation Approach to Normalized Hypergraph Cut

Input: Consider Input: We define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions (e.g., pi) and iterated connectives or ranging over intervals parameterized by arithmetic variables (e.g., n i=1 pi, where n is a parameter). The satisfiability problem is shown to be undecidable for this new logic, but we introduce a very general class of schemata, called bound-linear, for which this problem becomes decidable. This result is obtained by reduction to a particular class of schemata called regular, for which we provide a sound and complete terminating proof procedure. This schemata calculus (called stab) allows one to capture proof patterns corresponding to a large class of problems specified in propositional logic. We also show that the satisfiability problem becomes again undecidable for slight extensions of this class, thus demonstrating that bound-linear schemata represent a good compromise between expressivity and decidability.

Output: Decidability and Undecidability Results for Propositional Schemata

Input: Consider Input: We consider the task of aggregating beliefs of sev eral experts. We assume that these beliefs are rep resented as probability distributions. We argue that the evaluation of any aggregation technique depends on the semantic context of this task. We propose a framework, in which we assume that nature generates samples from a 'true' distribution and different experts form their beliefs based on the subsets of the data they have a chance to observe. Naturally, the optimal ag gregate distribution would be the one learned from the combined sample sets. Such a formulation leads to a natural way to measure the accuracy of the aggregation mechanism. We show that the well-known aggregation operator LinOP is ideally suited for that task. We propose a LinOP-based learning algorithm, inspired by the techniques developed for Bayesian learning, which aggregates the experts' distributions represented as Bayesian networks. We show experimentally that this algorithm performs well in practice.
Output: Aggregating Learned Probabilistic Beliefs