Smooth low-rank representation with a Grassmann manifold for tensor completion

Abstract

Utilizing low-rank representation, recent methods have efficiently estimated the low-rank tensor for tensor completion (TC). However, owing to the identifiability issue, these methods inevitably preserve some outliers of the sparse tensor as elements of the low-rank tensor, which leads to an imprecise estimation of the low-rank tensor. In this paper, we propose a smooth low-rank representation with a Grassmann manifold (SLRR-GM) model to tackle the identifiability issue. Specifically, we first smooth the low-frequency and high-frequency discontinuities of the low-rank tensor to sufficiently remove the preserved outliers of the sparse tensor. Accordingly, a smooth low-rank representation model is built to precisely estimate the low-rank tensor. Second, we smooth the discontinuities through a small number of the feature vectors that characterize the original vectors of the low-rank tensor. This uses the Grassmann manifold to effectively reduce the computational complexity of smoothing the discontinuities. Third, we integrate the smooth low-rank representation model with the Grassmann manifold to propose SLRR-GM. Based on the alternating direction method of multipliers (ADMM), an algorithm is proposed to solve the SLRR-GM model. We also investigate the complexity and convergence to validate the algorithmic effectiveness. Experiments in different scenarios demonstrate that the proposed method outperforms several state-of-the-art methods in terms of the quantitative and visual aspects with the acceptable processing time.