  Matrix valued data has become increasingly prevalent in many applications.
Most of the existing clustering methods for this type of data are tailored to
the mean model and do not account for the dependence structure of the features,
which can be very informative, especially in high-dimensional settings or when
mean information is not available. To extract the information from the
dependence structure for clustering, we propose a new latent variable model for
the features arranged in matrix form, with some unknown membership matrices
representing the clusters for the rows and columns. Under this model, we
further propose a class of hierarchical clustering algorithms using the
difference of a weighted covariance matrix as the dissimilarity measure.
Theoretically, we show that under mild conditions, our algorithm attains
clustering consistency in the high-dimensional setting. While this consistency
result holds for our algorithm with a broad class of weighted covariance
matrices, the conditions for this result depend on the choice of the weight. To
investigate how the weight affects the theoretical performance of our
algorithm, we establish the minimax lower bound for clustering under our latent
variable model in terms of some cluster separation metric. Given these results,
we identify the optimal weight in the sense that using this weight guarantees
our algorithm to be minimax rate-optimal. The practical implementation of our
algorithm with the optimal weight is also discussed. Simulation studies show
that our algorithm performs better than existing methods in terms of the
adjusted Rand index (ARI). The method is applied to a genomic dataset and
yields meaningful interpretations.
