Greedy randomized block Kaczmarz method for matrix equation AXB=C and its applications in color image restoration

TL;DR

This work develops and analyzes greedy randomized block Kaczmarz methods for the matrix equation $AXB=C$, establishing convergence to the least-norm solution $X_{*}=A^{+}CB^{+}$ (when consistent) and deriving faster convergence than prior ME-RBK approaches. It introduces ME-GRBK and its relaxation and deterministic variants (ME-RGRBK, ME-MWRBK), providing explicit convergence rates and conditions, and extends these ideas to practical applications. Theoretical results are complemented by numerical experiments showing superior performance in iteration counts and CPU time, and by a color image restoration application where the methods yield improved PSNR and SSIM over baseline approaches. Overall, the greedy randomized block strategy offers robust, scalable solutions for large-scale AXB=C problems with tangible benefits in image processing tasks.

Abstract

In view of the advantages of simplicity and effectiveness of the Kaczmarz method, which was originally employed to solve the large-scale system of linear equations $Ax=b$, we study the greedy randomized block Kaczmarz method (ME-GRBK) and its relaxation and deterministic versions to solve the matrix equation $AXB=C$, which is commonly encountered in the applications of engineering sciences. It is demonstrated that our algorithms converge to the unique least-norm solution of the matrix equation when it is consistent and their convergence rate is faster than that of the randomized block Kaczmarz method (ME-RBK). Moreover, the block Kaczmarz method (ME-BK) for solving the matrix equation $AXB=C$ is investigated and it is found that the ME-BK method converges to the solution $A^{+}CB^{+}+X^{0}-A^{+}AX^{0}BB^{+}$ when it is consistent. The numerical tests verify the theoretical results and the methods presented in this paper are applied to the color image restoration problem to obtain satisfactory restored images.

Figures (8)

• Figure 1: Convergence curves of RRN versus IT for ME-BK method for six set of coefficient matrices in Table 1.
• Figure 2: Convergence curves of RRN versus IT for ME-RBK, ME-GRBK, ME-RGRBK and ME-MWRBK methods for the 1-st set of coefficient matrices in Table 3.
• Figure 3: Convergence curves of RRN versus IT for ME-RBK, ME-GRBK, ME-RGRBK and ME-MWRBK methods for the 3-rd set of coefficient matrices in Table 5.
• Figure 4: Convergence curves of RRN versus IT for ME-RBK, ME-GRBK, ME-RGRBK and ME-MWRBK methods for the 2-nd set of coefficient matrices in Table 7.
• Figure 5: Image restoration results. Top row: test image (left), blurred image (middle), restored image by ME-RBK method (right). Bottom row: restored images by ME-GRBK (left), ME-RGRBK (middle) and ME-MWRBK (right) methods.

Theorems & Definitions (13)

• Lemma 2.1
• Lemma 2.2
• Remark 3.1
• Lemma 3.2
• Theorem 3.3
• Remark 4.1
• Theorem 4.2
• Remark 4.3
• Remark 4.4
• Theorem 4.5