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Tensor decomposition is a powerful tool for extracting physically meaningfullatent factors from multi-dimensional nonnegative data, and has been anincreasing interest in a variety of fields such as image processing, machinelearning, and computer vision. In this paper, we propose a sparse nonnegativeTucker decomposition and completion method for the recovery of underlyingnonnegative data under noisy observations. Here the underlying nonnegative datatensor is decomposed into a core tensor and several factor matrices with allentries being nonnegative and the factor matrices being sparse. The lossfunction is derived by the maximum likelihood estimation of the noisyobservations, and the $\ell_0$ norm is employed to enhance the sparsity of thefactor matrices. We establish the error bound of the estimator of the proposedmodel under generic noise scenarios, which is then specified to theobservations with additive Gaussian noise, additive Laplace noise, and Poissonobservations, respectively. Our theoretical results are better than those byexisting tensor-based or matrix-based methods. Moreover, the minimax lowerbounds are shown to be matched with the derived upper bounds up to logarithmicfactors. Numerical examples on both synthetic and real-world data setsdemonstrate the superiority of the proposed method for nonnegative tensor datacompletion.

Noisy Nonnegative Tucker Decomposition with Sparse Factors and Missing Data