Summarizing Boolean and fuzzy tensors with sub-tensors

The disjunctive box cluster model summarizes an n-way Boolean tensor with some of its sub-tensors and their densities, i.e., the arithmetic means of their values. Mirkin and Kramarenko proposed that easy-to-interpret regression model, for $n\in \{2,3\}$, and hill climbing to discover good sub-tensors, according to ordinary least squares. This article generalizes Mirkin and Kramarenko's work: n-way fuzzy tensors are summarized. They encode to what extent n-ary predicates are satisfied. The article also details significant performance improvements to the sequential execution, its parallelization, better starting points for hill climbing, a selection of the discovered sub-tensors, their ranking in order of contribution to the model, and the use of the elbow method to truncate the ordered list. In-depth experiments using synthetic and real-world tensors compare the proposed method, NclusterBox, to Mirkin and Kramarenko's and to the state-of-the-art algorithms for matrix factorization using the max (rather than +) operator and for Boolean tensor factorization. NclusterBox summarizes synthetic and real-world fuzzy tensors more efficiently and, most importantly, more accurately.