△-Closure Structure for Studying Data Distribution

In this paper, we revisit pattern mining and study the distribution underlying a binary dataset thanks to the closure structure which is based on passkeys, i.e., minimum generators in equivalence classes robust to noise. We introduce △-closedness, a generalization of the closure operator, where △ measures how a closed set differs from its upper neighbors in the partial order induced by closure. A △-class of equivalence includes minimum and maximum elements and allows us to characterize the distribution underlying the data. Moreover, the set of △-classes of equivalence can be partitioned into the so-called △-closure structure. In particular, a △-class of equivalence with a high △ is supported by more observations and thus is more stable. In the experiments, we study the △-closure structure of several real-world datasets and show that this structure is very stable for large △ and does not substantially depend on the data sampling used for the analysis.