  We consider the problem of jointly modeling and clustering populations of
tensors by introducing a high-dimensional tensor mixture model with
heterogeneous covariances. To effectively tackle the high dimensionality of
tensor objects, we employ plausible dimension reduction assumptions that
exploit the intrinsic structures of tensors such as low-rankness in the mean
and separability in the covariance. In estimation, we develop an efficient
high-dimensional expectation-conditional-maximization (HECM) algorithm that
breaks the intractable optimization in the M-step into a sequence of much
simpler conditional optimization problems, each of which is convex, admits
regularization and has closed-form updating formulas. Our theoretical analysis
is challenged by both the non-convexity in the EM-type estimation and having
access to only the solutions of conditional maximizations in the M-step,
leading to the notion of dual non-convexity. We demonstrate that the proposed
HECM algorithm, with an appropriate initialization, converges geometrically to
a neighborhood that is within statistical precision of the true parameter. The
efficacy of our proposed method is demonstrated through comparative numerical
experiments and an application to a medical study, where our proposal achieves
an improved clustering accuracy over existing benchmarking methods.
