Given a database and a target attribute of interest, how can we tell whether there exists a functional, or approximately functional dependence of the target on any set of other attributes in the data? How can we reliably, without bias to sample size or dimensionality, measure the strength of such a dependence? And, how can we efficiently discover the optimal or $\alpha$-approximate top-$k$ dependencies? These are exactly the questions we answer in this paper. As we want to be agnostic on the form of the dependence, we adopt an information-theoretic approach, and construct a reliable, bias correcting score that can be efficiently computed. Moreover, we give an effective optimistic estimator of this score, by which for the first time we can mine the approximate functional dependencies from data with guarantees of optimality. Empirical evaluation shows that the derived score achieves a good bias for variance trade-off, can be used within an efficient discovery algorithm, and indeed discovers meaningful dependencies. Most important, it remains reliable in the face of data sparsity.Comment: Accepted: In Proceedings of the ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD), August 13-17, 2017, Halifax, NS, Canad

Finding patterns from binary data is a classical problem in data mining, dating back to at least frequent itemset mining. More recently, approaches such as tiling and Boolean matrix factorization (BMF), have been proposed to find sets of patterns that aim to explain the full data well. These methods, however, are not robust against non-trivial destructive noise, i.e. when relatively many 1s are removed from the data: tiling can only model additive noise while BMF assumes approximately equal amounts of additive and destructive noise. Most real-world binary datasets, however, exhibit mostly destructive noise. In presence/absence data, for instance, it is much more common to fail to observe something than it is to observe a spurious presence. To address this problem, we take the recent approach of employing the Minimum Description Length (MDL) principle for BMF and introduce a new algorithm, Nassau, that directly optimizes the description length of the factorization instead of the reconstruction error. In addition, unlike the previous algorithms, it can adjust the factors it has discovered during its search. Empirical evaluation on synthetic data shows that Nassau excels at datasets with high destructive noise levels and its performance on real-world datasets confirms our hypothesis of the high numbers of missing observations in the real-world data

The reliable fraction of information is an attractive score for quantifying (functional) dependencies in highdimensional data. In this paper, we systematically explore the algorithmic implications of using this measure for optimization. We show that the problem is NP-hard, which justifies the usage of worst-case exponential-time as well as heuristic search methods. We then substantially improve the practical performance for both optimization styles by deriving a novel admissible bounding function that has an unbounded potential for additional pruning over the previously proposed one. Finally, we empirically investigate the approximation ratio of the greedy algorithm and show that it produces highly competitive results in a fraction of time needed for complete branch-and-bound style search.Index Terms-knowledge discovery, approximate functional dependency, information theory, optimization, branch-and-bound arXiv:1809.05467v1 [cs.AI]

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