Clustering Higher Order Data: Finite Mixtures of Multidimensional Arrays

There have been many examples of clustering multivariate (i.e., two-way) data using finite mixture models (see, e.g., reviews by Fraley and Raftery, 2002; Bouveyron and Brunet-Saumard, 2014; McNicholas, 2016b). More recently, there have been some notable examples of clustering threeway data using finite mixtures of matrix-variate distributions (e.g., Viroli, 2011; Anderlucci et al., 2015; Gallaugher and McNicholas, 2018a). This work on clustering three-way data is timely in the sense that the variety of data that require clustering continues to increase. Furthermore, there is no reason to believe that this need ends with three-way data. An approach for clustering multi-way data is introduced based on a finite mixture of multidimensional arrays. While some might refer to such structures as'tensors', and so write about clustering tensor-variate data, we prefer the nomenclature multidimensional array to avoid confusion with the term'tensor' as used in engineering and physics, e.g., tensor fields.