Robust Low-rank Tensor Completion based on Tensor Ring Rank via ℓp,ϵ-norm

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Original languageEnglish
Pages (from-to)3685-3698
Journal / PublicationIEEE Transactions on Signal Processing
Online published31 May 2021
Publication statusPublished - 2021


Tensor completion aims to recover missing entries given incomplete multi-dimensional data by making use of the prior low-rank information, and has various applications because many real-world data can be modeled as low-rank tensors. Most of the existing methods based on different tensor rank definitions are designed for noiseless or Gaussian noise scenarios, and thus they are not robust in the presence of outliers. One popular approach to resist outliers is to employ ℓp-norm. Yet nonsmoothness and nonconvexity of ℓp-norm with 0 < p ≤ 1 bring challenges to optimization. In this paper, a new norm, named ℓp,ϵ-norm, is devised where ϵ > 0 can adjust the convexity of ℓp,ϵ-norm. Compared with ℓp-norm, ℓp,ϵ-norm is smooth and convex even for 0 < p ≤ 1, which converts an intractable nonsmooth and nonconvex optimization problem into a much simpler convex and smooth one. Then, combining tensor ring rank and ℓp,ϵ-norm, a robust tensor completion formulation is proposed, which has a strong ability to fit various types of data and achieve outstanding robustness. The resultant robust tensor completion problem is decomposed into a number of robust linear regression (RLR) subproblems, and two algorithms are devised to tackle RLR. The first method adopts gradient descent, which has a low computational complexity. While the second one employs alternating direction method of multipliers to yield a fast convergence rate. Numerical simulations using synthetic data show that the two proposed methods have better performance than those based on the ℓp-norm in RLR. Experimental results from applications of color image inpainting, color video restoration and target estimation demonstrate that our robust tensor completion approach outperforms state-of-the-art methods in terms of recovery accuracy.

Research Area(s)

  • Tensor completion, tensor ring rank, linear regression, outlier, robust recovery, gradient descent, alternating direction method of multipliers