  Dimension reduction techniques are often used when the high-dimensional
tensor has relatively low intrinsic rank compared to the ambient dimension of
the tensor. The CANDECOMP/PARAFAC (CP) tensor completion is a widely used
approach to find a low-rank approximation for a given tensor. In the tensor
model, an $\ell_1$ regularized optimization problem was formulated with an
appropriate choice of the regularization parameter. The choice of the
regularization parameter is important in the approximation accuracy. However,
the emergence of the large amount of data poses onerous computational burden
for computing the regularization parameter via classical approaches such as the
weighted generalized cross validation (WGCV), the unbiased predictive risk
estimator, and the discrepancy principle. In order to improve the efficiency of
choosing the regularization parameter and leverage the accuracy of the CP
tensor, we propose a new algorithm for tensor completion by embedding the
flexible hybrid method into the framework of the CP tensor. The main benefits
of this method include incorporating regularization automatically and
efficiently, improved reconstruction and algorithmic robustness. Numerical
examples from image reconstruction and model order reduction demonstrate the
performance of the propose algorithm.
