Generalized Higher Order Orthogonal Iteration for Tensor Learning and Decomposition.

Abstract

Low-rank tensor completion (LRTC) has successfully been applied to a wide range of real-world problems. Despite the broad, successful applications, existing LRTC methods may become very slow or even not applicable for large-scale problems. To address this issue, a novel core tensor trace-norm minimization (CTNM) method is proposed for simultaneous tensor learning and decomposition, and has a much lower computational complexity. In our solution, first, the equivalence relation of trace norm of a low-rank tensor and its core tensor is induced. Second, the trace norm of the core tensor is used to replace that of the whole tensor, which leads to two much smaller scale matrix TNM problems. Finally, an efficient alternating direction augmented Lagrangian method is developed to solve our problems. Our CTNM formulation needs only O((RN+NRI)log(√{IN})) observations to reliably recover an N th-order I×I×…×I tensor of n -rank (r,r,…,r) , compared with O(rIN-1) observations required by those tensor TNM methods ( I >> R ≥ r ). Extensive experimental results show that CTNM is usually more accurate than them, and is orders of magnitude faster.