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Modern scientific studies often collect datasets in the form of tensors. These datasets call for innovative statistical analysis methods. In particular, there is a pressing need for tensor clustering methods to understand the heterogeneity in the data. We propose a tensor normal mixture model approach to enable probabilistic interpretation and computational tractability. Our statistical model leverages the tensor covariance structure to reduce the number of parameters for parsimonious modeling, and at the same time explicitly exploits the correlations for better variable selection and clustering. We propose a doubly enhanced expectation–maximization (DEEM) algorithm to perform clustering under this model. Both the expectation-step and the maximization-step are carefully tailored for tensor data in order to maximize statistical accuracy and minimize computational costs in high dimensions. Theoretical studies confirm that DEEM achieves consistent clustering even when the dimension of each mode of the tensors grows at an exponential rate of the sample size. Numerical studies demonstrate favorable performance of DEEM in comparison to existing methods.

A Doubly Enhanced EM Algorithm for Model-Based Tensor Clustering