Low-CP-rank Tensor Completion via Practical Regularization

Abstract

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.