  Mixtures of matrix Gaussian distributions provide a probabilistic framework
for clustering continuous matrix-variate data, which are becoming increasingly
prevalent in various fields. Despite its widespread adoption and successful
application, this approach suffers from over-parameterization issues, making it
less suitable even for matrix-variate data of moderate size. To overcome this
drawback, we introduce a sparse model-based clustering approach for three-way
data. Our approach assumes that the matrix mixture parameters are sparse and
have different degree of sparsity across clusters, allowing to induce parsimony
in a flexible manner. Estimation of the model relies on the maximization of a
penalized likelihood, with specifically tailored group and graphical lasso
penalties. These penalties enable the selection of the most informative
features for clustering three-way data where variables are recorded over
multiple occasions and allow to capture cluster-specific association
structures. The proposed methodology is tested extensively on synthetic data
and its validity is demonstrated in application to time-dependent crime
patterns in different US cities.
