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Abstract:
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We consider the problem of modeling heterogeneous tensor-valued data by jointly clustering and estimating heterogeneous tensor graphical models. We assume the data follow a mixture tensor normal distribution, whose mixture mean and covariances are unknown. Our formulation greatly relaxes the assumptions in existing approaches, which usually require that the cluster structure is given and/or the covariances are known in advance. To tackle this challenging high-dimensional problem, we effectively exploit the inherit structures of the tensor-valued data. Specifically, we assume the mixture mean is low-rank and the mixture covariance has a Kronecker structure. We formulate the parameter estimation as a conditional EM (CEM) estimation problem, and develop an efficient alternating minimization CEM algorithm. In theory, we establish the non-asymptotic error bound for the actual estimator from our algorithm. The theoretical analysis is highly nontrivial due to the non-convex nature of both the CEM estimation and tensor decomposition. The derived error bound reveals an interesting interplay between computational efficiency and the statistical rate of convergence.
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