  Higher-order singular value decomposition (HOSVD) is an efficient way for
data reduction and also eliciting intrinsic structure of multi-dimensional
array data. It has been used in many applications, and some of them involve
incomplete data. To obtain HOSVD of the data with missing values, one can first
impute the missing entries through a certain tensor completion method and then
perform HOSVD to the reconstructed data. However, the two-step procedure can be
inefficient and does not make reliable decomposition.
  In this paper, we formulate an incomplete HOSVD problem and combine the two
steps into solving a single optimization problem, which simultaneously achieves
imputation of missing values and also tensor decomposition. We also present two
algorithms for solving the problem based on block coordinate update. Global
convergence of both algorithms is shown under mild assumptions. The convergence
of the second algorithm implies that of the popular higher-order orthogonality
iteration (HOOI) method, and thus we, for the first time, give global
convergence of HOOI.
  In addition, we compare the proposed methods to state-of-the-art ones for
solving incomplete HOSVD and also low-rank tensor completion problems and
demonstrate the superior performance of our methods over other compared ones.
Furthermore, we apply them to face recognition and MRI image reconstruction to
show their practical performance.
