Rank-Revealing Bayesian Block-Term Tensor Completion With Graph Information

Block-term decomposition (BTD), particularly its rank-(Lr,Lr, 1) special case, is widely used in signal processing. Traditional methods for computing BTD either unrealistically assume the number of blocks and block ranks are known or require exhaustive tuning of these parameters. While sparsity-promoting regularization has been introduced to estimate these parameters more efficiently, it still requires regularization parameter tuning. Bayesian learning addresses these issues by employing sparsity-promoting priors on the number of blocks and block ranks, but so far is limited to fully observed BTD tensors. To process incomplete BTD tensors, only a few optimization-based methods have been proposed, and they continue to suffer from heavy parameter tuning. To enable tuning-free BTD completion, a prior that simultaneously enforces block-wise sparsity and withinblock column-wise sparsity while incorporating graph structure is introduced within the Bayesian framework. Besides theoretically establishing the legitimacy of the prior distribution, a mean-field design is developed to obtain a closed-form updating variational inference (VI) algorithm without loss of graph information. Extensive experiments on both synthetic datasets and real-world datasets demonstrate the superiority of the proposed method over existing optimization-based algorithms and the Bayesian model without graph information, in terms of rank learning, tensor recovery, and factor recovery.