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Prof. Dr.
Wrobel, Stefan
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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 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. 
PublicationDecision Snippet Features( 20210505)
;Welke, Pascal ;Alkhoury, FouadDecision trees excel at interpretability of their prediction results. To achieve required prediction accuracies, however, often large ensembles of decision trees random forests are considered, reducing interpretability due to large size. Additionally, their size slows down inference on modern hardware and restricts their applicability in lowmemory embedded devices. We introduce Decision Snippet Features, which are obtained from small subtrees that appear frequently in trained random forests. We subsequently show that linear models on top of these features achieve comparable and sometimes even better predictive performance than the original random forest, while reducing the model size by up to two orders of magnitude. 
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. 
PublicationA Novel Regression Loss for NonParametric Uncertainty Optimization( 2021)
;Sicking, Joachim ;Pintz, Maximilian ;Fischer, AsjaQuantification of uncertainty is one of the most promising approaches to establish safe machine learning. Despite its importance, it is far from being generally solved, especially for neural networks. One of the most commonly used approaches so far is Monte Carlo dropout, which is computationally cheap and easy to apply in practice. However, it can underestimate the uncertainty. We propose a new objective, referred to as secondmoment loss (SML), to address this issue. While the full network is encouraged to model the mean, the dropout networks are explicitly used to optimize the model variance. We intensively study the performance of the new objective on various UCI regression datasets. Comparing to the stateoftheart of deep ensembles, SML leads to comparable prediction accuracies and uncertainty estimates while only requiring a single model. Under distribution shift, we observe moderate improvements. As a side result, we introduce an intuitive Wasserstein distancebased uncertainty measure that is nonsaturating and thus allows to resolve quality differences between any two uncertainty estimates. 

PublicationHOPS: Probabilistic Subtree Mining for Small and Large Graphs( 2020)
;Welke, Pascal ;Seiffahrt, FlorianFrequent subgraph mining, i.e., the identification of relevant patterns in graph databases, is a wellknown data mining problem with high practical relevance, since next to summarizing the data, the resulting patterns can also be used to define powerful domainspecific similarity functions for prediction. In recent years, significant progress has been made towards subgraph mining algorithms that scale to complex graphs by focusing on tree patterns and probabilistically allowing a small amount of incompleteness in the result. Nonetheless, the complexity of the pattern matching component used for deciding subtree isomorphism on arbitrary graphs has significantly limited the scalability of existing approaches. In this paper, we adapt sampling techniques from mathematical combinatorics to the problem of probabilistic subtree mining in arbitrary databases of many small to mediumsize graphs or a single large graph. By restricting on tree patterns, we provide an algorithm tha t approximately counts or decides subtree isomorphism for arbitrary transaction graphs in sublinear time with onesided error. Our empirical evaluation on a range of benchmark graph datasets shows that the novel algorithm substantially outperforms stateoftheart approaches both in the task of approximate counting of embeddings in single large graphs and in probabilistic frequent subtree mining in large databases of small to medium sized graphs. 
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. 
PublicationAdiabatic Quantum Computing for MaxSum Diversification( 2020)The combinatorial problem of maxsum diversification asks for a maximally diverse subset of a given set of data. Here, we show that it can be expressed as an Ising energy minimization problem. Given this result, maxsum diversification can be solved on adiabatic quantum computers and we present proof of concept simulations which support this claim. This, in turn, suggests that quantum computing might play a role in data mining. We therefore discuss quantum computing in a tutorial like manner and elaborate on its current strengths and weaknesses for data analysis.