  We consider the problem of jointly modeling and clustering populations of
tensors by introducing a flexible high-dimensional tensor mixture model with
heterogeneous covariances. The proposed mixture model exploits the intrinsic
structures of tensor data, and is assumed to have means that are low-rank and
internally sparse as well as heterogeneous covariances that are separable and
conditionally sparse. We develop an efficient high-dimensional
expectation-conditional-maximization (HECM) algorithm that breaks the
challenging optimization in the M-step into several simpler conditional
optimization problems, each of which is convex, admits regularization and has
closed-form updating formulas. We show that the proposed HECM algorithm, with
an appropriate initialization, converges geometrically to a neighborhood that
is within statistical precision of the true parameter. Such a theoretical
analysis is highly nontrivial due to the dual non-convexity arising from both
the EM-type estimation and the non-convex objective function in the M-step. The
efficacy of our proposed method is demonstrated through simulation studies and
an application to an autism spectrum disorder study, where our analysis
identifies important brain regions for diagnosis.
