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<p>To efficiently solve consistent tensor linear systems based on the t-product, this paper first develops a tensor greedy randomized Kaczmarz (TGRK) method, which selects the horizontal slice with the largest residual norm for updating at each iteration and serves as a foundational framework. Building upon this, the paper proposes a tensor greedy randomized average block Kaczmarz method based on <i>k</i>-means clustering (TGRABK-kmeans). By partitioning via <i>k</i>-means clustering and employing an average block Kaczmarz update strategy, this method fully exploits structural information and significantly improves iteration efficiency. We prove the convergence of the proposed methods for consistent tensor linear systems. Experimental results demonstrate that TGRABK-kmeans exhibits satisfactory performance particularly in image reconstruction quality across multiple datasets, which validates the effectiveness and advantages of this method.</p>

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Greedy and Clustering-Enhanced Average Block Randomized Kaczmarz Methods for Solving Tensor Linear Systems

  • 期刊论文

  • 发表日期：

    2026年4月6日

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  • Yu-Hong Ran,
  • Yu-Shan Ma,
  • Jing Dong,
  • Jun-Gang Wang

摘要

To efficiently solve consistent tensor linear systems based on the t-product, this paper first develops a tensor greedy randomized Kaczmarz (TGRK) method, which selects the horizontal slice with the largest residual norm for updating at each iteration and serves as a foundational framework. Building upon this, the paper proposes a tensor greedy randomized average block Kaczmarz method based on k-means clustering (TGRABK-kmeans). By partitioning via k-means clustering and employing an average block Kaczmarz update strategy, this method fully exploits structural information and significantly improves iteration efficiency. We prove the convergence of the proposed methods for consistent tensor linear systems. Experimental results demonstrate that TGRABK-kmeans exhibits satisfactory performance particularly in image reconstruction quality across multiple datasets, which validates the effectiveness and advantages of this method.