Now showing 1 - 8 of 8
  • Publication
    A Simple Heuristic for the Graph Tukey Depth Problem with Potential Applications to Graph Mining
    ( 2022)
    Seiffarth, Florian
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    We 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 NP-hard 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 core-periphery 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.
  • Publication
    Learning Weakly Convex Sets in Metric Spaces
    ( 2021-09-10)
    Stadtländer, Eike
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    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.
  • Publication
    Mining Tree Patterns with Partially Injective Homomorphisms
    ( 2019)
    Schulz, Till Hendrik
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    Welke, Pascal
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    One 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.
  • Publication
    Probabilistic and exact frequent subtree mining in graphs beyond forests
    ( 2019)
    Welke, Pascal
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    Motivated 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.
  • Publication
    Support Estimation in Frequent Itemset Mining by Locality Sensitive Hashing
    The main computational effort in generating all frequent itemsets in a transactional database is in the step of deciding whether an itemset is frequent, or not. We present a method for estimating itemset supports with two-sided error. In a preprocessing step our algorithm first partitions the database into groups of similar transactions by using locality sensitive hashing and calculates a summary for each of these groups. The support of a query itemset is then estimated by means of these summaries. Our preliminary empirical results indicate that the proposed method results in a speed-up of up to a factor of 50 on large datasets. The F-measure of the output patterns varies between 0.83 and 0.99.
  • Publication
    Probabilistic frequent subtrees for efficient graph classification and retrieval
    ( 2018)
    Welke, Pascal
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    Frequent subgraphs proved to be powerful features for graph classification and prediction tasks. Their practical use is, however, limited by the computational intractability of pattern enumeration and that of graph embedding into frequent subgraph feature spaces. We propose a simple probabilistic technique that resolves both limitations. In particular, we restrict the pattern language to trees and relax the demand on the completeness of the mining algorithm, as well as on the correctness of the pattern matching operator by replacing transaction and query graphs with small random samples of their spanning trees. In this way we consider only a random subset of frequent subtrees, called probabilistic frequent subtrees, that can be enumerated efficiently. Our extensive empirical evaluation on artificial and benchmark molecular graph datasets shows that probabilistic frequent subtrees can be listed in practically feasible time and that their predictive and retrieval performance is very close even to those of complete sets of frequent subgraphs. We also present different fast techniques for computing the embedding of unseen graphs into (probabilistic frequent) subtree feature spaces. These algorithms utilize the partial order on tree patterns induced by subgraph isomorphism and, as we show empirically, require much less evaluations of subtree isomorphism than the standard brute-force algorithm. We also consider partial embeddings, i.e., when only a part of the feature vector has to be calculated. In particular, we propose a highly effective practical algorithm that significantly reduces the number of pattern matching evaluations required by the classical min-hashing algorithm approximating Jaccard-similarities.
  • Publication
    Listing closed sets of strongly accessible set systems with applications to data mining
    We study the problem of listing all closed sets of a closure operator a that is a partial function on the power set of some finite ground set E, i.e., sigma : F -> F with F subset of P(E). A very simple divide-and-conquer algorithm is analyzed that correctly solves this problem if and only if the domain of the closure operator is a strongly accessible set system. Strong accessibility is a strict relaxation of greedoids as well as of independence systems. This algorithm turns out to have delay O (vertical bar E vertical bar (T-F + T-sigma + vertical bar E vertical bar)) and space O (vertical bar E vertical bar + S-F + S-sigma), where T-F, S-F, T-sigma, and S-sigma are the time and space complexities of checking membership in F and computing a, respectively. In contrast, we show that the problem becomes intractable for accessible set systems. We relate our results to the data mining problem of listing all support-closed patterns of a dataset and show that there is a corresponding closure operator for all datasets if and only if the set system satisfies a certain confluence property.