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PublicationLearning Weakly Convex Sets in Metric Spaces( 2021-09-10)
;Stadtländer, Eike ;We introduce the notion of weak convexity in metric spaces, a generalization of ordinary convexity commonly used in machine learning. It is shown that weakly convex sets can be characterized by a closure operator and have a unique decomposition into a set of pairwise disjoint connected blocks. We give two generic efficient algorithms, an extensional and an intensional one for learning weakly convex concepts and study their formal properties. Our experimental results concerning vertex classification clearly demonstrate the excellent predictive performance of the extensional algorithm. Two non-trivial applications of the intensional algorithm to polynomial PAC-learnability are presented. The first one deals with learning k-convex Boolean functions, which are already known to be efficiently PAC-learnable. It is shown how to derive this positive result in a fairly easy way by the generic intensional algorithm. The second one is concerned with the Euclidean space equipped with the Manhattan distance. For this metric space, weakly convex sets form a union of pairwise disjoint axis-aligned hyperrectangles. We show that a weakly convex set that is consistent with a set of examples and contains a minimum number of hyperrectangles can be found in polynomial time. In contrast, this problem is known to be NP-complete if the hyperrectangles may be overlapping.
PublicationMining Tree Patterns with Partially Injective Homomorphisms( 2019)
;Schulz, Till Hendrik ; ;Welke, PascalOne of the main differences between inductive logic programming (ILP) and graph mining lies in the pattern matching operator applied: While it is mainly defined by relational homomorphism (i.e., subsumption) in ILP, subgraph isomorphism is the most common pattern matching operator in graph mining. Using the fact that subgraph isomorphisms are injective homomorphisms, we bridge the gap between ILP and graph mining by considering a natural transition from homomorphisms to subgraph isomorphisms that is defined by partially injective homomorphisms, i.e., which require injectivity only for subsets of the vertex pairs in the pattern. Utilizing positive complexity results on deciding homomorphisms from bounded tree-width graphs, we present an algorithm mining frequent trees from arbitrary graphs w.r.t. partially injective homomorphisms. Our experimental results show that the predictive performance of the patterns obtained is comparable to that of ordinary frequent subgraphs. Thus, by preserving much from the advantageous properties of homomorphisms and subgraph isomorphisms, our approach provides a trade-off between efficiency and predictive power.