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<p>This paper focuses on recovering a low-multilinear-rank tensor from its incomplete measurements. We propose a novel algorithm termed the Single-Mode Quasi Riemannian Gradient Descent (SM-QRGD) method. The SM-QRGD algorithm integrates the strengths of the fixed-rank matrix tangent space projection and the sequentially truncated high-order singular value decomposition (ST-HOSVD). This hybrid approach enables SM-QRGD to attain computational complexity per iteration of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(3 n^d r\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <msup> <mi>n</mi> <mi>d</mi> </msup> <mi>r</mi> </mrow> </math></EquationSource> </InlineEquation>, where <i>n</i> and <i>r</i> represent the tensor’s size and multilinear rank. This leads to a reduced computation cost per iteration, compared to other methods with the complexity coefficient related to the tensor order <i>d</i>. Theoretically, we establish the convergence of SM-QRGD through the Tensor Restricted Isometry Property (TRIP) and the structural properties of the fixed-rank matrix manifold. On the practical side, a comprehensive range of experiments validates the accuracy and efficacy of the proposed algorithm SM-QRGD for low-multilinear-rank tensor recovery.</p>

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A Single-Mode Quasi Riemannian Gradient Descent Algorithm for Low-Multilinear-Rank Tensor Recovery

  • 期刊论文

  • 发表日期：

    2025年12月9日

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  • Yuanwei Zhang,
  • Ya-Nan Zhu,
  • Xiaoqun Zhang

摘要

This paper focuses on recovering a low-multilinear-rank tensor from its incomplete measurements. We propose a novel algorithm termed the Single-Mode Quasi Riemannian Gradient Descent (SM-QRGD) method. The SM-QRGD algorithm integrates the strengths of the fixed-rank matrix tangent space projection and the sequentially truncated high-order singular value decomposition (ST-HOSVD). This hybrid approach enables SM-QRGD to attain computational complexity per iteration of \(3 n^d r\) 3 n d r , where n and r represent the tensor’s size and multilinear rank. This leads to a reduced computation cost per iteration, compared to other methods with the complexity coefficient related to the tensor order d. Theoretically, we establish the convergence of SM-QRGD through the Tensor Restricted Isometry Property (TRIP) and the structural properties of the fixed-rank matrix manifold. On the practical side, a comprehensive range of experiments validates the accuracy and efficacy of the proposed algorithm SM-QRGD for low-multilinear-rank tensor recovery.