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PublicationMaximum Margin Separations in Finite Closure SystemsMonotone linkage functions provide a measure for proximities between elements and subsets of a ground set. Combining this notion with Vapniks idea of support vector machines, we extend the concepts of maximal closed set and half-space separation in finite closure systems to those with maximum margin. In particular, we define the notion of margin for finite closure systems by means of monotone linkage functions and give a greedy algorithm computing a maximum margin closed set separation for two sets efficiently. The output closed sets are maximum margin half-spaces, i.e., form a partitioning of the ground set if the closure system is Kakutani. We have empirically evaluated our approach on different synthetic datasets. In addition to binary classification of finite subsets of the Euclidean space, we considered also the problem of vertex classification in graphs. Our experimental results provide clear evidence that maximal closed set separation with maximum margin results in a much better predictive performance than that with arbitrary maximal closed sets.
PublicationMaximal Closed Set and Half-Space Separations in Finite Closure SystemsMotivated by various binary classification problems in structured data (e.g., graphs or other relational and algebraic structures), we investigate some algorithmic properties of closed set and half-space separation in abstract closure systems. Assuming that the underlying closure system is finite and given by the corresponding closure operator, we formulate some negative and positive complexity results for these two separation problems. In particular, we prove that deciding half-space separability in abstract closure systems is NP-complete in general. On the other hand, for the relaxed problem of maximal closed set separation we propose a simple greedy algorithm and show that it is efficient and has the best possible lower bound on the number of closure operator calls. As a second direction to overcome the negative result above, we consider Kakutani closure systems and show first that our greedy algorithm provides an algorithmic characterization of this kind of set systems. As one of the major potential application fields, we then focus on Kakutani closure systems over graphs and generalize a fundamental characterization result based on the Pasch axiom to graph structure partitioning of finite sets. Though the primary focus of this work is on the generality of the results obtained, we experimentally demonstrate the practical usefulness of our approach on vertex classification in different graph datasets.