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PublicationA generalized Weisfeiler-Lehman graph kernel( 2022-04-27)
;Schulz, Till Hendrik ; ;Welke, PascalAfter 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.
PublicationProbabilistic and exact frequent subtree mining in graphs beyond forests( 2019)
;Welke, Pascal ;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.
PublicationProbabilistic frequent subtrees for efficient graph classification and retrieval( 2018)
;Welke, Pascal ;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.