# Delta-Closure Structure for Studying Data Distribution

arXiv: https://arxiv.org/abs/2210.06926 · HTML: https://mlsanity.com/paper/2210.06926/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 20 references that are in the map.
- First posted on arXiv 13 Oct 2022.

## Builds on (most cited first, 3 of 3)

- [On interestingness measures of formal concepts](https://mlsanity.com/paper/1611.02646/) (2016)
- [Flexible constrained sampling with guarantees for pattern mining](https://mlsanity.com/paper/1610.09263/) (2016)
- [Finding the True Frequent Itemsets](https://mlsanity.com/paper/1301.1218/) (2013)

## Abstract (arXiv)

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 Delta-closedness, a generalization of the closure operator, where Delta measures how a closed set differs from its upper neighbors in the partial order induced by closure. A Delta-class of equivalence includes minimum and maximum elements and allows us to characterize the distribution underlying the data. Moreover, the set of Delta-classes of equivalence can be partitioned into the so-called Delta-closure structure. In particular, a Delta-class of equivalence with a high level demonstrates correlations among many attributes, which are supported by more observations when Delta is large. In the experiments, we study the Delta-closure structure of several real-world datasets and show that this structure is very stable for large Delta and does not substantially depend on the data sampling used for the analysis.