Tensor Completion

The purpose of this thesis is to explore the methods to solve the tensor completion problem. Inspired by the matrix completion problem, the tensor completion problem is formulated as an unconstrained nonlinear optimization problem, which finds three factors that give a low-rank approximation. Various of iterative methods, including the gradient-based methods, stochastic gradient descent method and the alternating least squares method, were applied to solving tensor completion problem, and were analyzed in case of taking the squared loss with L2 regularization as the loss function. Aiming at the large-scale problem, the author proposed for each of these methods a parallelization strategy. Numerical experiments using synthetic data and real world benchmark demonstrated the speedup of the parallelization, and compared the relative performance as well as the scalability of these methods. The results suggests that both stochastic gradient descent method and alternating least squares method have advantages in solving the large-scale tensor completion problem.