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Wrobel, Stefan
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PublicationMaximal closed set and halfspace separations in finite closure systems( 20230921)
;Seiffarth, FlorianSeveral concept learning problems can be regarded as special cases of halfspace separation in abstract closure systems over finite ground sets. For the typical scenario that the closure system is given via a closure operator, we show that the halfspace separation problem is NPcomplete. As a first approach to overcome this negative result, we relax the problem to maximal closed set separation, give a simple generic greedy algorithm solving this problem with a linear number of closure operator calls, and show that this bound is sharp. For a second direction, we consider Kakutani closure systems and prove that they are algorithmically characterized by the greedy algorithm. As a first special case of the general problem setting, we consider Kakutani closure systems over graphs and give a sufficient condition for this kind of closure systems in terms of forbidden graph minors. For a second special case, we then focus on closure systems over finite lattices, give an improved adaptation of the generic greedy algorithm, and present an application concerning subsumption lattices. 
PublicationA Fast Heuristic for Computing Geodesic Closures in Large Networks( 20221106)
;Seiffarth, FlorianMotivated by the increasing interest in applications of graph geodesic convexity in machine learning and data mining, we present a heuristic for approximating the geodesic convex hull of node sets in large networks. It generates a small set of (almost) maximal outerplanar spanning subgraphs for the input graph, computes the geodesic closure in each of these graphs, and regards a node as an element of the convex hull if it belongs to the closed sets for at least a user specified number of outerplanar graphs. Our heuristic algorithm runs in time linear in the number of edges of the input graph, i.e., it is faster with one order of magnitude than the standard algorithm computing the closure exactly. Its performance is evaluated empirically by approximating convexity based coreperiphery decomposition of networks. Our experimental results with large realworld networks show that for most networks, the proposed heuristic was able to produce close approximations significantly faster than the standard algorithm computing the exact convex hulls. For example, while our algorithm calculated an approximate coreperiphery decomposition in 5 h or less for networks with more than 20 million edges, the standard algorithm did not terminate within 50 days. 
PublicationA generalized WeisfeilerLehman graph kernel( 20220427)
;Schulz, Till Hendrik ;Welke, PascalAfter more than one decade, WeisfeilerLehman graph kernels are still among the most prevalent graph kernels due to their remarkable predictive performance and time complexity. They are based on a fast iterative partitioning of vertices, originally designed for deciding graph isomorphism with onesided error. The WeisfeilerLehman graph kernels retain this idea and compare such labels with respect to equality. This binary valued comparison is, however, arguably too rigid for defining suitable graph kernels for certain graph classes. To overcome this limitation, we propose a generalization of WeisfeilerLehman graph kernels which takes into account a more natural and finer grade of similarity between WeisfeilerLehman labels than equality. We show that the proposed similarity can be calculated efficiently by means of the Wasserstein distance between certain vectors representing WeisfeilerLehman labels. This and other facts give rise to the natural choice of partitioning the vertices with the Wasserstein kmeans algorithm. We empirically demonstrate on the WeisfeilerLehman subtree kernel, which is one of the most prominent WeisfeilerLehman graph kernels, that our generalization significantly outperforms this and other stateoftheart graph kernels in terms of predictive performance on datasets which contain structurally more complex graphs beyond the typically considered molecular graphs. 
PublicationA Simple Heuristic for the Graph Tukey Depth Problem with Potential Applications to Graph Mining( 2022)
;Seiffarth, FlorianWe study a recently introduced adaptation of Tukey depth to graphs and discuss its algorithmic properties and potential applications to mining and learning with graphs. In particular, since it is NPhard to compute the Tukey depth of a node, as a first contribution we provide a simple heuristic based on maximal closed set separation in graphs and show empirically on different graph datasets that its approximation error is small. Our second contribution is concerned with geodesic coreperiphery decompositions of graphs. We show empirically that the geodesic core of a graph consists of those nodes that have a high Tukey depth. This information allows for a parameterized deterministic definition of the geodesic core of a graph. 
PublicationLearning Weakly Convex Sets in Metric Spaces( 20210910)
;Stadtländer, EikeWe 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 nontrivial applications of the intensional algorithm to polynomial PAClearnability are presented. The first one deals with learning kconvex Boolean functions, which are already known to be efficiently PAClearnable. 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 axisaligned 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 NPcomplete if the hyperrectangles may be overlapping. 
PublicationMaximum Margin Separations in Finite Closure Systems( 2021)
;Seiffahrt, FlorianMonotone 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 halfspace 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 halfspaces, 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 HalfSpace Separations in Finite Closure Systems( 2020)
;Seiffarth, FlorianMotivated 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 halfspace 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 halfspace separability in abstract closure systems is NPcomplete 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. 
PublicationEffective approximation of parametrized closure systems over transactional data streams( 2020)Strongly closed itemsets, defined by a parameterized closure operator, are a generalization of ordinary closed itemsets. Depending on the strength of closedness, the family of strongly closed itemsets typically forms a tiny subfamily of ordinary closed itemsets that is stable against changes in the input. In this paper we consider the problem of mining strongly closed itemsets from transactional data streams. Utilizing their algebraic and algorithmic properties, we propose an algorithm based on reservoir sampling for approximating this type of itemsets in the landmark streaming setting, prove its correctness, and show empirically that it yields a considerable speedup over a straightforward naive algorithm without any significant loss in precision and recall. We motivate the problem setting considered by two practical applications. In particular, we first experimentally demonstrate that the above properties, i.e., compactness and stability, make strongly closed itemsets an excellent indicator of certain types of concept drifts in transactional data streams. As a second application we consider computeraided product configuration, a realworld problem raised by an industrial project. For this problem, which is essentially exact concept identification, we propose a learning algorithm based on a certain type of subset queries formed by strongly closed itemsets and show on realworld datasets that it requires significantly less query evaluations than a naive algorithm based on membership queries.

PublicationProbabilistic and exact frequent subtree mining in graphs beyond forests( 2019)
;Welke, PascalMotivated by the impressive predictive power of simple patterns, we consider the problem of mining frequent subtrees in arbitrary graphs. Although the restriction of the pattern language to trees does not resolve the computational complexity of frequent subgraph mining, in a recent work we have shown that it gives rise to an algorithm generating probabilistic frequent subtrees, a random subset of all frequent subtrees, from arbitrary graphs with polynomial delay. It is based on replacing each transaction graph in the input database with a forest formed by a random subset of its spanning trees. This simple technique turned out to be quite powerful on molecule classification tasks. It has, however, the drawback that the number of sampled spanning trees must be bounded by a polynomial of the size of the transaction graphs, resulting in less impressive recall even for slightly more complex structures beyond molecular graphs. To overcome this limitation, in this work we propose an algorithm mining probabilistic frequent subtrees also with polynomial delay, but by replacing each graph with a forest formed by an exponentially large implicit subset of its spanning trees. We demonstrate the superiority of our algorithm over the simple one on threshold graphs used e.g. in spectral clustering. In addition, providing sufficient conditions for the completeness and efficiency of our algorithm, we obtain a positive complexity result on exact frequent subtree mining for a novel, practically and theoretically relevant graph class that is orthogonal to all graph classes defined by some constant bound on monotone graph properties. 
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 treewidth 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 tradeoff between efficiency and predictive power.