  Boolean tensor decomposition approximates data of multi-way binary
relationships as product of interpretable low-rank binary factors, following
the rules of Boolean algebra. Here, we present its first probabilistic
treatment. We facilitate scalable sampling-based posterior inference by
exploitation of the combinatorial structure of the factor conditionals. Maximum
a posteriori decompositions feature higher accuracies than existing techniques
throughout a wide range of simulated conditions. Moreover, the probabilistic
approach facilitates the treatment of missing data and enables model selection
with much greater accuracy. We investigate three real-world data-sets. First,
temporal interaction networks in a hospital ward and behavioural data of
university students demonstrate the inference of instructive latent patterns.
Next, we decompose a tensor with more than 10 billion data points, indicating
relations of gene expression in cancer patients. Not only does this demonstrate
scalability, it also provides an entirely novel perspective on relational
properties of continuous data and, in the present example, on the molecular
heterogeneity of cancer. Our implementation is available on GitHub:
https://github.com/TammoR/LogicalFactorisationMachines.
