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Quaternion tensor low rank Quaternion tensor low-rank approximation using a family of non-convex norms

Alaeddine Zahir, Ahmed Ratnani, Khalide Jbilou

TL;DR

The paper addresses low-rank recovery and denoising of high-dimensional color data modeled as quaternion tensors. It develops non-convex surrogate penalties for Tucker- and TT-rank notions within the QT-product framework and solves the resulting problems with ADMM-based proximal updates, providing convergence guarantees. The authors introduce LRQTC-NCTR, LRQTC-NCTTR, and TRPCA-NC, and validate their superiority over convex nuclear-norm baselines on color video inpainting and denoising tasks. The work demonstrates robust quaternion-tensor recovery with promising extensions to higher-dimensional algebras (e.g., Octonions) and potential for parallel computation.

Abstract

In this paper, we propose a new approaches for low rank approximation of quaternion tensors \cite{chen2019low,zhang1997quaternions,hamilton1866elements}. The first method uses quasi-norms to approximate the tensor by a low-rank tensor using the QT-product \cite{miao2023quaternion}, which generalizes the known L-product to N-mode quaternions. The second method involves Non-Convex norms to approximate the Tucker and TT-rank for the completion problem. We demonstrate that the proposed methods can effectively approximate the tensor compared to the convexifying of the rank, such as the nuclear norm. We provide theoretical results and numerical experiments to show the efficiency of the proposed methods in the Inpainting and Denoising applications.

Quaternion tensor low rank Quaternion tensor low-rank approximation using a family of non-convex norms

TL;DR

The paper addresses low-rank recovery and denoising of high-dimensional color data modeled as quaternion tensors. It develops non-convex surrogate penalties for Tucker- and TT-rank notions within the QT-product framework and solves the resulting problems with ADMM-based proximal updates, providing convergence guarantees. The authors introduce LRQTC-NCTR, LRQTC-NCTTR, and TRPCA-NC, and validate their superiority over convex nuclear-norm baselines on color video inpainting and denoising tasks. The work demonstrates robust quaternion-tensor recovery with promising extensions to higher-dimensional algebras (e.g., Octonions) and potential for parallel computation.

Abstract

In this paper, we propose a new approaches for low rank approximation of quaternion tensors \cite{chen2019low,zhang1997quaternions,hamilton1866elements}. The first method uses quasi-norms to approximate the tensor by a low-rank tensor using the QT-product \cite{miao2023quaternion}, which generalizes the known L-product to N-mode quaternions. The second method involves Non-Convex norms to approximate the Tucker and TT-rank for the completion problem. We demonstrate that the proposed methods can effectively approximate the tensor compared to the convexifying of the rank, such as the nuclear norm. We provide theoretical results and numerical experiments to show the efficiency of the proposed methods in the Inpainting and Denoising applications.
Paper Structure (17 sections, 9 theorems, 43 equations, 3 figures, 2 tables, 4 algorithms)

This paper contains 17 sections, 9 theorems, 43 equations, 3 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic quaternion algebra and notations
  4. Quaternion matrix and tensor
  5. The QT-product
  6. Introducing the Non-Convex surrogate functions
  7. Low rank tensor completion
  8. Solution of the proposed model
  9. Convergence analysis
  10. Low rank tensor problem
  11. Solution of the proposed model
  12. Extension work
  13. Experiments
  14. Completion task
  15. Denoising task
  16. ...and 2 more sections

Key Result

Theorem 2.1

Given $\dot{Q} \in \mathbb{H}^{m \times n}$, there exist two unitary matrices $\dot{U} \in \mathbb{H}^{m \times m},\dot{V} \in \mathbb{H}^{n \times n}$ and a real rectangular diagonal matrix $S=\operatorname{diag}(\sigma_i)$, such that The decomposition is called quaternion singular value decomposition (QSVD).

Figures (3)

  • Figure 6.1: Visual results of different methods on different Sample rate $[0.1,0.5]$, on resized $48\times 72$ colored chosen frame image from Pedestrian Dataset. From left to right columns: Noised Frame, Tucker-nc, Ttucker, Ttucker-nc and TTmac.
  • Figure 6.2: PSNR and SSIM results on different frames of Highway Dataset, with $SR=0.5$
  • Figure 6.3: Visual results of different methods on Noise level $\%$, on a frame image from Highway Dataset. The forth and fifth column, are TRPCA-NC-rand, and TRPCA-NC-dct, respectively.

Theorems & Definitions (23)

  • Theorem 2.1: QSVD zhang1997quaternions
  • Definition 2.2
  • Definition 2.3: Mode-k unfolding
  • Definition 2.4: Tucker Rank
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8: QT-SVD
  • Definition 2.9
  • Definition 2.10
  • ...and 13 more