  Finite Gaussian mixture models provide a powerful and widely employed
probabilistic approach for clustering multivariate continuous data. However,
the practical usefulness of these models is jeopardized in high-dimensional
spaces, where they tend to be over-parameterized. As a consequence, different
solutions have been proposed, often relying on matrix decompositions or
variable selection strategies. Recently, a methodological link between Gaussian
graphical models and finite mixtures has been established, paving the way for
penalized model-based clustering in the presence of large precision matrices.
Notwithstanding, current methodologies implicitly assume similar levels of
sparsity across the classes, not accounting for different degrees of
association between the variables across groups. We overcome this limitation by
deriving group-wise penalty factors, which automatically enforce under or
over-connectivity in the estimated graphs. The approach is entirely data-driven
and does not require additional hyper-parameter specification. Analyses on
synthetic and real data showcase the validity of our proposal.
