  In this paper, we study the sparse nonnegative tensor factorization and
completion problem from partial and noisy observations for third-order tensors.
Because of sparsity and nonnegativity, the underlying tensor is decomposed into
the tensor-tensor product of one sparse nonnegative tensor and one nonnegative
tensor. We propose to minimize the sum of the maximum likelihood estimation for
the observations with nonnegativity constraints and the tensor $\ell_0$ norm
for the sparse factor. We show that the error bounds of the estimator of the
proposed model can be established under general noise observations. The
detailed error bounds under specific noise distributions including additive
Gaussian noise, additive Laplace noise, and Poisson observations can be
derived. Moreover, the minimax lower bounds are shown to be matched with the
established upper bounds up to a logarithmic factor of the sizes of the
underlying tensor. These theoretical results for tensors are better than those
obtained for matrices, and this illustrates the advantage of the use of
nonnegative sparse tensor models for completion and denoising. Numerical
experiments are provided to validate the superiority of the proposed
tensor-based method compared with the matrix-based approach.
