The Efficient Discovery of Interesting Closed Pattern Collections

Enumerating closed sets that are frequent in a given database is a fundamental data mining technique that is used, e.g., in the context of market basket analysis, fraud detection, or Web personalization. There are two complementing reasons for the importance of closed sets— one semantical and one algorithmic: closed sets provide a condensed basis for non-redundant collections of interesting local patterns, and they can be enumerated efficiently. For many databases, however, even the closed set collection can be way too large for further usage and correspondingly its computation time can be infeasibly long. In such cases, it is inevitable to focus on smaller collections of closed sets, and it is essential that these collections retain both: controlled semantics reflecting some notion of interestingness as well as efficient enumerability. This thesis discusses three different approaches to achieve this: constraint-based closed set extraction, pruning by quantifying the degree or strength of closedness, and controlled random generation of closed sets instead of exhaustive enumeration. For the original closed set family, efficient enumerability results from the fact that there is an inducing efficiently computable closure operator and that its fixpoints can be enumerated by an amortized polynomial number of closure computations. Perhaps surprisingly, it turns out that this connection does not generally hold for other constraint combinations, as the restricted domains induced by additional constraints can cause two things to happen: the fixpoints of the closure operator cannot be enumerated efficiently or an inducing closure operator does not even exist. This thesis gives, for the first time, a formal axiomatic characterization of constraint classes that allow to efficiently enumerate fixpoints of arbitrary closure operators as well as of constraint classes that guarantee the existence of a closure operator inducing the closed sets. As a complementary approach, the thesis generalizes the notion of closedness by quantifying its strength, i.e., the difference in supporting database records between a closed set and all its supersets. This gives rise to a measure of interestingness that is able to select long and thus particularly informative closed sets that are robust against noise and dynamic changes. Moreover, this measure is algorithmically sound because all closed sets with a minimum strength again form a closure system that can be enumerated efficiently and that directly ties into the results on constraint-based closed sets. In fact both approaches can easily be combined. In some applications, however, the resulting set of constrained closed sets is still intractably large or it is too difficult to find meaningful hard constraints at all (including values for their parameters). Therefore, the last part of this thesis presents an alternative algorithmic paradigm to the extraction of closed sets: instead of exhaustively listing a potentially exponential number of sets, randomly generate exactly the desired amount of them. By using the Markov chain Monte Carlo method, this generation can be performed according to any desired probability distribution that favors interesting patterns. This novel randomized approach complements traditional enumeration techniques (including those mentioned above): On the one hand, it is only applicable in scenarios that do not require deterministic guarantees for the output such as exploratory data analysis or global model construction. On the other hand, random closed set generation provides complete control over the number as well as the distribution of the produced sets.