Now showing 1 - 9 of 9
  • Publication
    Maximal closed set and half-space separations in finite closure systems
    ( 2023-09-21)
    Seiffarth, Florian
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    Several concept learning problems can be regarded as special cases of half-space 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 half-space separation problem is NP-complete. 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.
  • Publication
    A generalized Weisfeiler-Lehman graph kernel
    ( 2022-04-27)
    Schulz, Till Hendrik
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    Welke, Pascal
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    After more than one decade, Weisfeiler-Lehman 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 one-sided error. The Weisfeiler-Lehman 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 Weisfeiler-Lehman graph kernels which takes into account a more natural and finer grade of similarity between Weisfeiler-Lehman labels than equality. We show that the proposed similarity can be calculated efficiently by means of the Wasserstein distance between certain vectors representing Weisfeiler-Lehman labels. This and other facts give rise to the natural choice of partitioning the vertices with the Wasserstein k-means algorithm. We empirically demonstrate on the Weisfeiler-Lehman subtree kernel, which is one of the most prominent Weisfeiler-Lehman graph kernels, that our generalization significantly outperforms this and other state-of-the-art graph kernels in terms of predictive performance on datasets which contain structurally more complex graphs beyond the typically considered molecular graphs.
  • Publication
    Effective approximation of parametrized closure systems over transactional data streams
    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 speed-up 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 computer-aided product configuration, a real-world 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 real-world datasets that it requires significantly less query evaluations than a naive algorithm based on membership queries.
  • 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
    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
    Frequent subgraph mining in outerplanar graphs
    In recent years there has been an increased interest in frequent pattern discovery in large databases of graph structured objects. While the frequent connected subgraph mining problem for tree datasets can be solved in incremental polynomial time, it becomes intractable for arbitrary graph databases. Existing approaches have therefore resorted to various heuristic strategies and restrictions of the search space, but have not identified a practically relevant tractable graph class beyond trees. In this paper, we consider the class of outerplanar graphs, a strict generalization of trees, develop a frequent subgraph mining algorithm for outerplanar graphs, and show that it works in incremental polynomial time for the practically relevant subclass of well-behaved outerplanar graphs, i.e., which have only polynomially many simple cycles. We evaluate the algorithm empirically on chemo- and bioinformatics applications.
  • 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.
  • Publication
    Efficient discovery of interesting patterns based on strong closedness
    Finding patterns that are interesting to a user in a certain application context is one of the central goals of data mining research. Regarding all patterns above a certain frequency threshold as interesting is one way of defining interestingness. In this paper, however, we argue that in many applications, a different notion of interestingness is required in order to be able to capture "long", and thus particularly informative, patterns that are correspondingly of low frequency. To identify such patterns, our proposed measure of interestingness is based on the degree or strength of closedness of the patterns. We show that (i) indeed this definition selects long interesting patterns that are difficult to identify with frequency-based approaches, and (ii) that it selects patterns that are robust against noise and/or dynamic changes. We prove that the family of interesting patterns proposed here forms a closure system and use the corresponding closure operator to design a mining algorithm listing these patterns in amortized quadratic time. In particular, for nonsparse datasets its time complexity is O(nm) per pattern, where n denotes the number of items and m the size of the database. This is equal to the best known time bound for listing ordinary closed frequent sets, which is a special case of our problem. We also report empirical results with real-world datasets.
  • Publication
    Support-vector-machine-based ranking significantly improves the effectiveness of similarity searching using 2D fingerprints and multiple reference compounds
    ( 2008)
    Geppert, H.
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    Bajorath, J.
    Similarity searching using molecular fingerprints is computationally efficient and a surprisingly effective virtual screening tool. In this study, we have compared ranking methods for similarity searching using multiple active reference molecules. Different 2D fingerprints were used as search tools and also as descriptors for a support vector machine (SVM) algorithm. In systematic database search calculations, a SVM-based ranking scheme consistently outperformed nearest neighbor and centroid approaches, regardless of the fingerprints that were tested, even if only very small training sets were used for SVM learning. The superiority of SVM-based ranking over conventional fingerprint methods is ascribed to the fact that SVM makes use of information about database molecules, in addition to known active compounds, during the learning phase.