💻 computer science

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

Original authors: Wenli Wang, Duo Liu, Gangrong Qu

Published 2026-02-05

📖 4 min read☕ Coffee break read

Original authors: Wenli Wang, Duo Liu, Gangrong Qu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a massive, tangled knot of strings. In the world of mathematics and engineering, this "knot" is a giant matrix equation (specifically $AXB = C$). Solving this equation is like trying to find the perfect arrangement of strings to match a specific target pattern. This problem shows up everywhere, from fixing blurry photos to analyzing complex data in machine learning.

For decades, mathematicians have used a tool called the Kaczmarz method to untangle these knots. Think of the classic Kaczmarz method as a very diligent, but slightly slow, worker who checks the strings one by one in a strict order (Row 1, then Row 2, then Row 3...). It works, but for huge knots, it takes forever.

This paper introduces a new, smarter team of workers to solve these equations faster. Here is how they work, explained simply:

1. The Old Way vs. The New "Greedy" Team

The authors propose three new methods: ME-GRBK, ME-RGRBK, and ME-MWRBK.

The Old Way (ME-RBK): Imagine a worker who picks a string to check completely at random. Sometimes they pick a string that is already straight (wasting time), and sometimes they pick a string that is very tangled (helpful). It's a bit of a gamble.
The New "Greedy" Way (ME-GRBK): This worker is "greedy" in a good way. Before picking a string, they look at the whole knot and ask, "Which string is the most messed up right now?" They prioritize the worst tangles. By focusing on the biggest problems first, they untangle the knot much faster.
The "Relaxed" Way (ME-RGRBK): This is like the greedy worker but with a little more flexibility. Sometimes, looking only at the worst string might be too rigid. This worker uses a "relaxation factor" (a dial they can turn) to decide how strictly to follow the "worst string" rule. It allows them to be smart but adaptable.
The "Deterministic" Way (ME-MWRBK): This is the most decisive worker. They don't gamble at all. They simply find the single most tangled string and fix it immediately. It's a "pick the worst and fix it" approach, guaranteed to be very efficient.

2. The "Block" Strategy

The paper also mentions a "Block" method. Imagine instead of fixing one string at a time, your worker grabs a whole bundle of strings (a block) and fixes them all at once.

The authors proved that if you use this "Block" method (ME-BK), you will eventually reach a solution. However, if you start with a messy guess, the final result might be slightly shifted from the "perfect" center.
The "Greedy" versions (GRBK, RGRBK, MWRBK) are even better. They not only use the bundle strategy but also pick the best bundles to fix, ensuring they reach the unique, perfect center (the "least-norm solution") of the knot, no matter where they started.

3. The "Color Image" Test

To prove these new workers are actually better, the authors tested them on a real-world task: restoring color images.

The Problem: Imagine taking a photo of a bird, but it gets blurred and noisy (like looking through a dirty window). The goal is to reverse the blur and get the clear bird back.
The Math: This restoration process is mathematically the same as solving that giant matrix equation ($AXB = C$).
The Result: The authors ran a race between the old random worker (ME-RBK) and their new greedy team.
- Speed: The new greedy methods finished the job much faster (using less computer time).
- Quality: The pictures restored by the new methods were sharper and looked more like the original bird. The "Peak Signal-to-Noise Ratio" (a fancy way of saying "how clear the picture is") was significantly higher for the new methods.

Summary of the Paper's Claims

The Problem: Solving huge matrix equations is hard and slow with old methods.
The Solution: The authors created three new "Greedy Randomized Block Kaczmarz" methods. They are like workers who intelligently pick the biggest problems to fix first, rather than guessing randomly.
The Proof: They mathematically proved these new methods will always find the correct answer (converge) and do it faster than the previous best method.
The Application: They tested this on color image restoration. The new methods cleaned up blurry photos better and faster than the old method.

In a nutshell: If you have a giant, messy puzzle, don't just pick pieces randomly. Look for the messiest pieces first, fix them, and you'll solve the puzzle much faster and with a better result. That is exactly what this paper teaches us how to do.

Technical Summary: Greedy Randomized Block Kaczmarz Method for Matrix Equation $AXB = C$ and Applications in Color Image Restoration

Problem Statement
The paper addresses the solution of large-scale matrix equations of the form $AXB = C$, where $A \in \mathbb{R}^{m \times p}$ , $B \in \mathbb{R}^{q \times n}$ , and $C \in \mathbb{R}^{m \times n}$ . This equation arises frequently in engineering applications, including image processing, stability analysis, control theory, and machine learning regression. While direct methods are impractical for large-scale systems, traditional iterative methods (e.g., gradient-based, Jacobi, Gauss-Seidel) often require storing and computing with the entire coefficient matrix at each step, creating high storage and computational demands. The authors focus on row-action and column-action methods, specifically extending the Kaczmarz method, which avoids full matrix access by operating on sub-blocks.

Methodology
The authors propose a suite of iterative algorithms based on the Kaczmarz framework, adapted for the matrix equation $AXB = C$:

Block Kaczmarz (ME-BK): A deterministic cyclic method where the row index $i_k$ is selected sequentially ( $i_k = (k \mod m) + 1$ ). The update projects the current estimate onto the hyperplane defined by the selected row of $A$ .
Greedy Randomized Block Kaczmarz (ME-GRBK): A stochastic method that selects rows based on a probability criterion favoring those with larger residuals. Specifically, it defines a threshold $\zeta_k$ based on the maximum normalized residual and the global residual norm. Rows satisfying $\|R_k^{i,:}\|_2^2 \ge \zeta_k \|A_{i,:}\|_2^2 \|R_k\|_F^2$ form a candidate set $J_k$ , from which a row is chosen with probability proportional to its squared residual.
Relaxed Greedy Randomized Block Kaczmarz (ME-RGRBK): An extension of ME-GRBK introducing a relaxation factor $\theta \in (0, 1)$ into the probability criterion to allow for more flexible index set selection.
Maximal Weighted Residual Block Kaczmarz (ME-MWRBK): A deterministic version of ME-GRBK that selects the row index maximizing the weighted residual $\frac{\|R_k^{i,:}\|_2^2}{\|A_{i,:}\|_2^2}$ .

The algorithms utilize the Kronecker product structure implicitly (via vectorization) and update the residual matrix $R_k = C - AX_kB$ efficiently by precomputing $AA^T$ and $B^TB$ .

Key Contributions

Convergence of ME-BK: The paper establishes that the deterministic ME-BK method converges to the solution $X^* + X_0 - A^+AX_0BB^+$ (where $X^* = A^+CB^+$ is the unique least-norm solution) when the system is consistent. This fills a gap, as the convergence of the block Kaczmarz method for $AXB=C$ had not been previously investigated.
Convergence of Greedy Variants: The authors prove that the ME-GRBK, ME-RGRBK, and ME-MWRBK methods converge to the unique least-norm solution $A^+CB^+$ in expectation when the system is consistent.
Convergence Rates: Theoretical analysis demonstrates that the convergence factors for the greedy and relaxed methods are strictly smaller (indicating faster convergence) than that of the existing Randomized Block Kaczmarz (ME-RBK) method presented in [12].
Distinction from Prior Work: The authors clarify that their approach differs from the method in [11]. While [11] transforms the equation into $mn$ subsystems requiring two stochastic indices, this paper transforms it into $m$ subsystems ( $A_{i,:}XB = C_{i,:}$ ) requiring only one stochastic index per iteration.

Results
Numerical experiments were conducted using MATLAB on various matrix sets (sparse and dense, full rank and rank-deficient) from the University of Florida collection and random generators.

Convergence Verification: Theoretical convergence of ME-BK to the specific solution form was verified across six matrix sets.
Performance Comparison: In terms of Iteration count (IT) and CPU time, the proposed ME-GRBK, ME-RGRBK, and ME-MWRBK methods consistently outperformed the ME-RBK method.
- For full column/row rank systems, speed-ups ranged from approximately 2x to 13x depending on the matrix set and method.
- For rank-deficient systems, speed-ups ranged from 1.35x to 4.29x.
- The ME-MWRBK (deterministic) generally achieved the highest speed-ups, followed closely by ME-RGRBK and ME-GRBK.
Application to Image Restoration: The methods were applied to color image restoration (deblurring) modeled as $B = AXA_c^T + E$ . Using test images ("face", "bird", "mandril", "barbara"), the proposed methods achieved significantly higher Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) compared to ME-RBK. For instance, on the "face" image, ME-RBK yielded a PSNR of 27.02, while ME-MWRBK achieved 33.72.

Significance
The paper claims that the proposed methods offer a more efficient and effective approach for solving large-scale matrix equations $AXB=C$ compared to existing randomized block Kaczmarz methods. By introducing greedy selection strategies and deterministic variants, the authors demonstrate superior convergence rates both theoretically and numerically. The successful application to color image restoration validates the practical utility of these algorithms in real-world engineering problems where large-scale data and storage constraints are critical factors. The work expands the applicability of Kaczmarz-type methods beyond linear systems to general matrix equations.

1. The Old Way vs. The New "Greedy" Team

2. The "Block" Strategy

3. The "Color Image" Test

Summary of the Paper's Claims

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