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Low-rank tensor completion via tensor joint rank with logarithmic composite norm

Hongbing Zhang

TL;DR

This work tackles low-rank tensor completion under severely limited observations by introducing the tensor joint rank (TJ) and a tensor logarithmic composite norm (LC) to jointly model Tucker and tubal ranks. An ADMM-style TJLC algorithm is developed, leveraging a proximal LC operator and data-fidelity updates, with theoretical convergence guarantees to a stationary point. Empirical results on MRI, MSI, and CV datasets show TJLC consistently surpasses state-of-the-art Tucker- and t-SVD-based methods, achieving accurate recovery even at observed information as low as one percent. The method thus substantially enhances information utilization in LRTC, with broad implications for high-dimensional data completion in imaging and vision tasks.

Abstract

Low-rank tensor completion (LRTC) aims to recover a complete low-rank tensor from incomplete observed tensor, attracting extensive attention in various practical applications such as image processing and computer vision. However, current methods often perform well only when there is a sufficient of observed information, and they perform poorly or may fail when the observed information is less than 5\%. In order to improve the utilization of observed information, a new method called the tensor joint rank with logarithmic composite norm (TJLC) method is proposed. This method simultaneously exploits two types of tensor low-rank structures, namely tensor Tucker rank and tubal rank, thereby enhancing the inherent correlations between known and missing elements. To address the challenge of applying two tensor ranks with significantly different directly to LRTC, a new tensor Logarithmic composite norm is further proposed. Subsequently, the TJLC model and algorithm for the LRTC problem are proposed. Additionally, theoretical convergence guarantees for the TJLC method are provided. Experiments on various real datasets demonstrate that the proposed method outperforms state-of-the-art methods significantly. Particularly, the proposed method achieves satisfactory recovery even when the observed information is as low as 1\%, and the recovery performance improves significantly as the observed information increases.

Low-rank tensor completion via tensor joint rank with logarithmic composite norm

TL;DR

This work tackles low-rank tensor completion under severely limited observations by introducing the tensor joint rank (TJ) and a tensor logarithmic composite norm (LC) to jointly model Tucker and tubal ranks. An ADMM-style TJLC algorithm is developed, leveraging a proximal LC operator and data-fidelity updates, with theoretical convergence guarantees to a stationary point. Empirical results on MRI, MSI, and CV datasets show TJLC consistently surpasses state-of-the-art Tucker- and t-SVD-based methods, achieving accurate recovery even at observed information as low as one percent. The method thus substantially enhances information utilization in LRTC, with broad implications for high-dimensional data completion in imaging and vision tasks.

Abstract

Low-rank tensor completion (LRTC) aims to recover a complete low-rank tensor from incomplete observed tensor, attracting extensive attention in various practical applications such as image processing and computer vision. However, current methods often perform well only when there is a sufficient of observed information, and they perform poorly or may fail when the observed information is less than 5\%. In order to improve the utilization of observed information, a new method called the tensor joint rank with logarithmic composite norm (TJLC) method is proposed. This method simultaneously exploits two types of tensor low-rank structures, namely tensor Tucker rank and tubal rank, thereby enhancing the inherent correlations between known and missing elements. To address the challenge of applying two tensor ranks with significantly different directly to LRTC, a new tensor Logarithmic composite norm is further proposed. Subsequently, the TJLC model and algorithm for the LRTC problem are proposed. Additionally, theoretical convergence guarantees for the TJLC method are provided. Experiments on various real datasets demonstrate that the proposed method outperforms state-of-the-art methods significantly. Particularly, the proposed method achieves satisfactory recovery even when the observed information is as low as 1\%, and the recovery performance improves significantly as the observed information increases.
Paper Structure (14 sections, 4 theorems, 44 equations, 5 figures, 5 tables, 1 algorithm)

This paper contains 14 sections, 4 theorems, 44 equations, 5 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensor joint rank and logarithmic composite norm
  4. TJLC model and solving algorithm
  5. Convergence analysis
  6. Experiments
  7. Experimental setup
  8. MRI data
  9. MSI data
  10. CV data
  11. Parameters setting
  12. Convergence behavior
  13. Discussion
  14. Conclusion and future work

Key Result

Theorem 1

Let $\mathcal{X}\in\mathbb{R}^{\mathit{I}_{1}\times\mathit{I}_{2}\times\mathit{I}_{3}}$ be a third-order tensor, then it can be factored as where $\mathcal{U}\in\mathbb{R}^{\mathit{I}_{1}\times\mathit{I}_{1}\times\mathit{I}_{3}}$ and $\mathcal{V}\in\mathbb{R}^{\mathit{I}_{2}\times\mathit{I}_{2}\times\mathit{I}_{3}}$ are orthogonal tensors, and $\mathcal{S}\in\mathbb{R}^{\mathit{I}_{1}\times\mathi

Figures (5)

  • Figure 1: Visual results for MRI data. MR: top row is 95%, and last row is 90%. The corresponding slices in each row are: 50, 100.
  • Figure 2: Visual results for MSI data. The rows of MSIs are in order: clay, chart_and_stuffed_toy, balloons, cd. MR: top two rows are 95%, and last two rows are 90%. The corresponding bands in each row are: 15, 20, 25, 30.
  • Figure 3: Visual results for CV data. The rows of CVs are in order: akiyo, hall, foreman, news, highway, container. MR: top three rows are 95%, and last three rows are 90%. The corresponding bands in each row are: 5, 15, 20, 40, 30, 45.
  • Figure 4: The convergence behaviours of TJLC algorithm for MRI, MSI and CV datas.
  • Figure 5: Visual results for MRI, MSI, and CV datas.

Theorems & Definitions (14)

  • definition 1: Tensor mode-$n$ unfolding and folding 12345152009
  • definition 2: Tensor Tucker rank 12345152009
  • definition 3: t-product 6416568
  • Theorem 1: t-SVD 8606166
  • definition 4: Tensor tubal-rank doi:10.1137/110837711
  • definition 5: Tensor mode-$l_{1}l_2$ unfolding and folding
  • definition 6: Tensor joint rank
  • definition 7: Tensor logarithmic composite norm
  • Theorem 2: Proximal operator for tensor logarithmic composite norm
  • proof
  • ...and 4 more