Triple Decomposition and Tensor Recovery of Third Order Tensors
Liqun Qi, Yannan Chen, Mayank Bakshi, Xinzhen Zhang
Abstract
Motivated by the Tucker decomposition, in this paper we introduce a new tensor decomposition for third order tensors, which decomposes a third order tensor to three third order factor tensors. Each factor tensor has two low dimensions. We call such a decomposition the triple decomposition, and the corresponding rank the triple rank. The triple rank of a third order tensor is not greater than the middle value of the Tucker rank. The number of parameters in the bilevel form of standard triple decomposition is less than the number of parameters of Tucker decomposition in substantial cases. The theoretical discovery is confirmed numerically. Numerical tests show that third order tensor data from practical applications such as internet traffic and image are of low triple ranks. A tensor recovery method based on low rank triple decomposition is proposed. Its convergence and convergence rate are established. Numerical experiments confirm the efficiency of this method.