High-order tensor completion via gradient-based optimization under tensor train format

Longhao Yuan, Qibin Zhao, Lihua Gui, Jianting Cao

Abstract: Highlights • By employing tensor-train (TT) decomposition, we propose a gradientbased tensor completion algorithm named TT-WOPT which is of low computational complexity and shows high computational efficiency. • Based on stochastic gradient descent method, we propose the TT-SGD algorithm which possesses extremely low computational complexity in every iteration and can be applied to solving large-scale tensor completion problems. • We propose a higher-order tensorization method named VDT which transforms visual data into higher-order tensors. By applying the VDT method, the performance of TT-WOPT and TT-SGD are improved. Abstract Tensor train (TT) decomposition has drawn people’s attention due to its powerful representation ability and performance stability in high-order tensors. In this paper, we propose a novel approach to recover the missing entries of incomplete data represented by higher-order tensors. We attempt to find the low-rank TT decomposition of the incomplete data which captures the latent features of the whole data and then reconstruct the missing entries. By applying gradient descent algorithms, tensor completion problem is efficiently solved by optimization models. We propose two TT-based algorithms: Tensor Train Weighted Optimization (TT-WOPT) and Tensor Train Stochastic Gradient Descent (TT-SGD) to optimize TT decomposition factors. In addition, a method named Visual Data Tensorization (VDT) is proposed to transform visual data into higher-order tensors, resulting in the performance improvement of our algorithms. The experiments in synthetic data and visual data show high efficiency and performance of our algorithms compared to the state-of-the-art completion algorithms, especially in high-order, high missing rate, and large-scale tensor completion situations.