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Handling The Non-Smooth Challenge in Tensor SVD: A Multi-Objective Tensor Recovery Framework

Jingjing Zheng, Wanglong Lu, Wenzhe Wang, Yankai Cao, Xiaoqin Zhang, Xianta Jiang

TL;DR

This work tackles the non‑smooth challenge and slice permutation variability that hinder traditional t‑SVD‑based tensor recovery in visual data. It introduces a learnable tensor nuclear norm framework, parameterized by unitary transforms, and optimizes it with an Alternating Proximal Multiplier Method (APMM), proving convergence to a KKT point. The authors further extend to a multi‑objective setting (MOTC) to jointly learn cross‑dimensional low‑rankness, solved via an APMM‑based heuristic and NSGA‑II, enabling scalable exploration across all mode pairs. Empirical results on image and color video inpainting demonstrate that TC‑SL and especially MOTC outperform state‑of‑the‑art methods, with notable PSNR gains under non‑smooth content changes and strong potential for application to higher‑order tensor tasks.

Abstract

Recently, numerous tensor singular value decomposition (t-SVD)-based tensor recovery methods have shown promise in processing visual data, such as color images and videos. However, these methods often suffer from severe performance degradation when confronted with tensor data exhibiting non-smooth changes. It has been commonly observed in real-world scenarios but ignored by the traditional t-SVD-based methods. In this work, we introduce a novel tensor recovery model with a learnable tensor nuclear norm to address such a challenge. We develop a new optimization algorithm named the Alternating Proximal Multiplier Method (APMM) to iteratively solve the proposed tensor completion model. Theoretical analysis demonstrates the convergence of the proposed APMM to the Karush-Kuhn-Tucker (KKT) point of the optimization problem. In addition, we propose a multi-objective tensor recovery framework based on APMM to efficiently explore the correlations of tensor data across its various dimensions, providing a new perspective on extending the t-SVD-based method to higher-order tensor cases. Numerical experiments demonstrated the effectiveness of the proposed method in tensor completion.

Handling The Non-Smooth Challenge in Tensor SVD: A Multi-Objective Tensor Recovery Framework

TL;DR

This work tackles the non‑smooth challenge and slice permutation variability that hinder traditional t‑SVD‑based tensor recovery in visual data. It introduces a learnable tensor nuclear norm framework, parameterized by unitary transforms, and optimizes it with an Alternating Proximal Multiplier Method (APMM), proving convergence to a KKT point. The authors further extend to a multi‑objective setting (MOTC) to jointly learn cross‑dimensional low‑rankness, solved via an APMM‑based heuristic and NSGA‑II, enabling scalable exploration across all mode pairs. Empirical results on image and color video inpainting demonstrate that TC‑SL and especially MOTC outperform state‑of‑the‑art methods, with notable PSNR gains under non‑smooth content changes and strong potential for application to higher‑order tensor tasks.

Abstract

Recently, numerous tensor singular value decomposition (t-SVD)-based tensor recovery methods have shown promise in processing visual data, such as color images and videos. However, these methods often suffer from severe performance degradation when confronted with tensor data exhibiting non-smooth changes. It has been commonly observed in real-world scenarios but ignored by the traditional t-SVD-based methods. In this work, we introduce a novel tensor recovery model with a learnable tensor nuclear norm to address such a challenge. We develop a new optimization algorithm named the Alternating Proximal Multiplier Method (APMM) to iteratively solve the proposed tensor completion model. Theoretical analysis demonstrates the convergence of the proposed APMM to the Karush-Kuhn-Tucker (KKT) point of the optimization problem. In addition, we propose a multi-objective tensor recovery framework based on APMM to efficiently explore the correlations of tensor data across its various dimensions, providing a new perspective on extending the t-SVD-based method to higher-order tensor cases. Numerical experiments demonstrated the effectiveness of the proposed method in tensor completion.
Paper Structure (18 sections, 1 theorem, 16 equations, 5 figures, 4 tables, 2 algorithms)

This paper contains 18 sections, 1 theorem, 16 equations, 5 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Multi-Objective Tensor Recovery with Learnable Tensor Nuclear Norms
  3. Tensor Completion Problem
  4. The Proposed Tensor Completion with A Learnable Tensor Rank Function
  5. Approximation to The Proposed Tensor Completion by Using A Learnable Tensor Nuclear Norm (TC-SL)
  6. The Proposed Multi-Objective Tensor Completion to Learn The Cross-Dimensional Low-Rankness
  7. Optimization Algorithm
  8. Alternating Proximal Multiplier Method (APMM) for TC-SL
  9. Computational Complexity
  10. Convergence Analysis
  11. APMM-based Heuristic Method for Solving MOTC
  12. Experimental Results
  13. Images Inpainting
  14. Image Sequences With Various Scenes
  15. Image Sequences With Random Shuffling
  16. ...and 3 more sections

Key Result

theorem 1

For the sequence $\{[\boldsymbol{\mathcal{Z}}^{(t)},\{\boldsymbol{U}_{k_n}^{(t)}\}_{n=s+1}^h, \boldsymbol{\mathcal{E}}^{(t)}, \boldsymbol{\mathcal{Y}}^{(t)},\mu^{(t)}]\}$ generated by the proposed algorithm Alg2, we have the following properties if $\{\boldsymbol{\mathcal{Y}}^{(t)}\}$ is bounded, $\

Figures (5)

  • Figure 1: Illustration to challenges of t-SVD-based methods in real world scenarios.
  • Figure 2: Examples of images inpainting by different methods on the BSD dataset with sampling rate $p=0.3$. Best viewed in $\times 2$ sized color pdf file.
  • Figure 3: Examples of images inpainting by different methods on three dataset with sampling rate $p=0.3$. Best viewed in $\times 2$ sized color pdf file.
  • Figure 4: Comparing PSNR by different methods on the 50 video segments at a sampling rate $p=0.3$.
  • Figure 5: Examples of video inpainting by different methods on HMDB51 dataset for case of $p=0.3$. Best viewed in $\times 2$ sized color pdf file.

Theorems & Definitions (2)

  • definition 1
  • theorem 1