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<p>The unique structure of quaternion tensors enables them to effectively capture and reflect correlations across diverse signal channels and modalities. This characteristic makes them invaluable in various scientific and engineering fields. However, research on quaternion tensors faces several challenges, particularly in the areas of high-order scalability of quaternion tensor operations and the rapid low-rank approximation of high-order quaternion tensor singular value decomposition (QtSVD). In this paper, we introduce a scalable QR decomposition for high-order quaternion tensors based on adaptive unitary quaternion transforms, designed for efficient computation of high-order quaternion tensor tri-factorization as an approximation of QtSVD. Additionally, to assess the significance of the obtained decomposition in low-rank regularization modeling, we define the nuclear norm and the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3132_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(L_{2,1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>-norm of high-order quaternion tensors. We then propose algorithms for color video inpainting, and present experiments using real data to demonstrate the effectiveness and efficiency of the proposed methods.</p>

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Quaternion tensor tri-factorization for the low-rank approximation with application to video inpainting

  • 期刊论文

  • 发表日期：

    2025年2月12日

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  • Fengsheng Wu,
  • Yonghe Liu,
  • Chaoqian Li

摘要

The unique structure of quaternion tensors enables them to effectively capture and reflect correlations across diverse signal channels and modalities. This characteristic makes them invaluable in various scientific and engineering fields. However, research on quaternion tensors faces several challenges, particularly in the areas of high-order scalability of quaternion tensor operations and the rapid low-rank approximation of high-order quaternion tensor singular value decomposition (QtSVD). In this paper, we introduce a scalable QR decomposition for high-order quaternion tensors based on adaptive unitary quaternion transforms, designed for efficient computation of high-order quaternion tensor tri-factorization as an approximation of QtSVD. Additionally, to assess the significance of the obtained decomposition in low-rank regularization modeling, we define the nuclear norm and the \(L_{2,1}\) L 2 , 1 -norm of high-order quaternion tensors. We then propose algorithms for color video inpainting, and present experiments using real data to demonstrate the effectiveness and efficiency of the proposed methods.