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

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.

💡 Research Summary

The paper addresses the computational challenge of solving large‑scale matrix equations of the form AXB = C, where A ∈ ℝ^{m×p}, B ∈ ℝ^{q×n} and C ∈ ℝ^{m×n}. Classical Kaczmarz methods, originally designed for linear systems Ax = b, have been extended to matrix equations, but existing randomized and block variants (e.g., ME‑RBK) select rows solely based on the norm of the rows of A, ignoring the current residual. This can limit practical convergence speed.

The authors propose a family of greedy randomized block Kaczmarz algorithms (ME‑GRBK) and two derived versions: a relaxed greedy version (ME‑RGRBK) and a maximal‑weighted‑residual version (ME‑MWRBK). The key idea is to decompose the matrix equation into m subsystems A_{i,:} X B = C_{i,:} and, at each iteration, select the block whose residual r_i = C_{i,:} − A_{i,:} X_k B has the largest Euclidean norm. The selection probability is proportional to ‖A_{i,:}‖·‖r_i‖, which biases the algorithm toward the most “informative’’ rows. The relaxed version introduces a scalar α (0 < α < 2‖B‖_2²) to smooth the probability distribution, while the maximal‑weighted‑residual version directly weights the probability by the product of row norm and residual norm.

The paper also revisits the deterministic block Kaczmarz method (ME‑BK), where rows are visited cyclically. The authors prove that ME‑BK converges to the solution X⁰* = A⁺ C B⁺ + X⁰ − A⁺ A X⁰ B B⁺, i.e., the least‑norm solution plus a correction term that depends on the initial guess. This result clarifies the behavior of block Kaczmarz in the consistent case, contrasting with the classical Kaczmarz convergence to the unique least‑norm solution.

The convergence analysis for all proposed methods relies on standard matrix inequalities (Lemma 2.1) linking the smallest singular values of A and B to their Frobenius norms, and on expectation arguments over the random block selection. For the greedy algorithms, the authors derive a linear convergence rate ρ = 1 − 2α − α²‖B‖_2²‖A‖_F²/(σ_min(A)²σ_min(B)²), which is strictly smaller than the rate for ME‑RBK, confirming faster theoretical convergence. The relaxed version allows tuning α to balance aggressiveness and stability, while the maximal‑weighted‑residual variant further accelerates convergence by emphasizing rows with large residuals.

Extensive numerical experiments are presented. Synthetic tests with random large matrices demonstrate that ME‑GRBK reduces the number of iterations needed to reach a tolerance of 10⁻⁶ by roughly 30–40 % compared with ME‑RBK, and also yields lower CPU times. In a practical application, the authors formulate color image restoration as a matrix equation where A models spatial blurring and B models down‑sampling. Applying the proposed algorithms to degraded images, they achieve higher PSNR (by about 1.5 dB) and SSIM scores than both ME‑RBK and traditional iterative schemes such as Jacobi or Gauss‑Seidel. Visual inspection confirms that edges and color details are better preserved, illustrating the practical benefit of the greedy block selection.

In conclusion, the paper contributes a rigorously analyzed set of greedy block Kaczmarz methods that outperform existing randomized block approaches both theoretically and empirically. The work opens avenues for further research, including extensions to non‑square block structures, nonlinear operators, and distributed/parallel implementations for real‑time image processing tasks.