A Tensor Greedy Double-Block Extended Kaczmarz Method for Inconsistent Tensor Linear Systems under the t-product

Abstract

The randomized extended Kaczmarz method is an effective iterative framework for solving large-scale inconsistent linear systems. In this paper, we extend this framework to third-order inconsistent tensor linear systems under the t-product and propose the Tensor Greedy Double Block Extended Kaczmarz (TGDBEK) method. At each iteration, TGDBEK dynamically constructs active blocks of row and column slices via a residual-based greedy selection strategy, prioritizing the slices associated with the largest residual norms. Unlike existing tensor block Kaczmarz variants that rely on static, predefined partitions --the tensor randomized extended block Kaczmarz (TREBK) method, its greedy counterpart (TREGBK), and the tensor randomized extended average block Kaczmarz (TREABK) method -- TGDBEK adapts the active block sizes dynamically at each step using a single intuitive threshold parameter $\eta$. We establish the theoretical linear convergence of TGDBEK to the unique minimum-norm least-squares solution $\mathcal{A}^\dagger * \mathcal{B}$. Extensive numerical benchmarks on synthetic dense and sparse tensor systems, as well as multidimensional multichannel color and 3D volumetric MRI image deblurring problems, demonstrate that TGDBEK substantially outperforms state-of-the-art tensor Kaczmarz solvers in both iteration count and CPU running time.