[Paper Review] Truncated nuclear norm regularization for low-rank tensor completion

Shengke Xue, Wenyuan Qiu+2 more|arXiv (Cornell University)|Jan 7, 2019

Sparse and Compressive Sensing Techniques27 references4 citations

TL;DR

This paper proposes a novel tensor truncated nuclear norm (T-TNN) for low-rank tensor completion, extending the matrix-level truncated nuclear norm to tensors using tensor singular value decomposition (t-SVD). By minimizing the sum of the smallest singular values across frontal slices in the Fourier domain, the method improves low-rank approximation accuracy and accelerates convergence. Experiments show T-TNN outperforms existing methods in both image and video recovery, achieving higher PSNR and faster convergence with ADMM and APGL solvers.

ABSTRACT

Recently, low-rank tensor completion has become increasingly attractive in recovering incomplete visual data. Considering a color image or video as a three-dimensional (3D) tensor, existing studies have put forward several definitions of tensor nuclear norm. However, they are limited and may not accurately approximate the real rank of a tensor, and they do not explicitly use the low-rank property in optimization. It is proved that the recently proposed truncated nuclear norm (TNN) can replace the traditional nuclear norm, as an improved approximation to the rank of a matrix. In this paper, we propose a new method called the tensor truncated nuclear norm (T-TNN), which suggests a new definition of tensor nuclear norm. The truncated nuclear norm is generalized from the matrix case to the tensor case. With the help of the low rankness of TNN, our approach improves the efficacy of tensor completion. We adopt the definition of the previously proposed tensor singular value decomposition, the alternating direction method of multipliers, and the accelerated proximal gradient line search method in our algorithm. Substantial experiments on real-world videos and images illustrate that the performance of our approach is better than those of previous methods.

Motivation & Objective

• To address the limitations of existing tensor nuclear norms in approximating tensor rank accurately.
• To improve low-rank tensor completion by generalizing the truncated nuclear norm from matrices to tensors.
• To exploit the low-rank structure more explicitly in optimization, enhancing recovery performance.
• To develop an efficient algorithm using t-SVD and accelerated solvers (ADMM, APGL) for faster convergence.
• To demonstrate superior performance on real-world image and video datasets compared to state-of-the-art methods.

Proposed method

• Proposes a new tensor nuclear norm definition based on the truncated nuclear norm, minimizing only the smallest singular values across frontal slices in the t-SVD framework.
• Utilizes tensor singular value decomposition (t-SVD) to decompose the tensor into Fourier-transformed frontal slices, enabling singular value operations.
• Applies the alternating direction method of multipliers (ADMM) and accelerated proximal gradient line search (APGL) to solve the optimization problem efficiently.
• Implements a truncated nuclear norm regularization that ignores the largest r singular values, improving rank approximation over standard nuclear norm.
• Solves the low-rank tensor completion problem via constrained minimization: minimize T-TNN subject to observed entries matching the original data.
• Employs a soft-thresholding-like update in the Fourier domain to handle the non-convex but tighter rank approximation.

Experimental results

Research questions

• RQ1 Can the truncated nuclear norm from matrix completion be effectively generalized to the tensor case for improved low-rank approximation?
• RQ2 Does using the t-SVD framework enable better tensor nuclear norm definition than existing methods like SMNN or ATNN?
• RQ3 How does the truncated nuclear norm improve convergence speed and recovery accuracy in tensor completion compared to standard nuclear norm or Tubal-NN?
• RQ4 Can the proposed T-TNN method achieve better PSNR and faster convergence than state-of-the-art tensor completion algorithms?
• RQ5 How sensitive is the performance to the choice of the number of truncated singular values (r), and can it be made more robust?

Key findings

• The proposed T-TNN method achieves higher PSNR than all compared methods, including LRMC, TNNR, SMNN, ATNN, and Tubal-NN, on real-world image and video datasets with 50% missing entries.
• T-TNN with ADMM and APGL solvers converges significantly faster than Tubal-NN and LRMC, with APGL requiring fewer iterations than ADMM.
• The running time of T-TNN methods is consistently under 40 seconds per image, outperforming Tubal-NN and LRMC in speed while maintaining high accuracy.
• Visual results show that T-TNN ADMM and APGL produce clearer reconstructions with fewer artifacts than LRMC, TNNR, SMNN, and ATNN, and are visually competitive with Tubal-NN.
• The SMNN and ATNN methods perform worse than LRMC in some cases, indicating their unsuitability as tensor nuclear norm definitions.
• The Tubal-NN method performs better than SMNN and ATNN but is still outperformed by the proposed T-TNN in both PSNR and convergence speed.