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Deterministic and randomized Kaczmarz methods for $AXB=C$ with applications to color image restoration

Wenli Wang, Duo Liu, Gangrong Qu, Michiel E. Hochstenbach

TL;DR

The paper addresses solving the consistent matrix equation $A X B = C$ efficiently for large-scale problems, including color image restoration models. It develops deterministic block Kaczmarz (BK) methods and randomized greedy block variants (GRBK, RGRBK, MWRBK), proving convergence to the minimum-norm solution $X_* = A^+ C B^+$ under mild conditions and providing cycle/sweep formulas in special full-rank cases. Theoretical convergence guarantees are complemented by extensive numerical experiments showing that greedy variants typically converge faster in iterations and wall-clock time than RBK or BK, with manageable costs. The color image restoration application demonstrates the methods' practicality for realistic multi-channel inverse problems and highlights the favorable performance of the deterministic maximal-weighted-residual (MWRBK) variant in practice.

Abstract

We study Kaczmarz type methods to solve consistent linear matrix equations. We first present a block Kaczmarz (BK) method that employs a deterministic cyclic row selection strategy. Assuming that the associated coefficient matrix has full column or row rank, we derive matrix formulas for a cycle of this BK method. Moreover, we propose a greedy randomized block Kaczmarz (GRBK) method and further extend it to a relaxed variant (RGRBK) and a deterministic counterpart (MWRBK). We establish the convergence properties of the proposed methods. Numerical tests verify the theoretical findings, and we apply the proposed methods to color image restoration problems.

Deterministic and randomized Kaczmarz methods for $AXB=C$ with applications to color image restoration

TL;DR

The paper addresses solving the consistent matrix equation efficiently for large-scale problems, including color image restoration models. It develops deterministic block Kaczmarz (BK) methods and randomized greedy block variants (GRBK, RGRBK, MWRBK), proving convergence to the minimum-norm solution under mild conditions and providing cycle/sweep formulas in special full-rank cases. Theoretical convergence guarantees are complemented by extensive numerical experiments showing that greedy variants typically converge faster in iterations and wall-clock time than RBK or BK, with manageable costs. The color image restoration application demonstrates the methods' practicality for realistic multi-channel inverse problems and highlights the favorable performance of the deterministic maximal-weighted-residual (MWRBK) variant in practice.

Abstract

We study Kaczmarz type methods to solve consistent linear matrix equations. We first present a block Kaczmarz (BK) method that employs a deterministic cyclic row selection strategy. Assuming that the associated coefficient matrix has full column or row rank, we derive matrix formulas for a cycle of this BK method. Moreover, we propose a greedy randomized block Kaczmarz (GRBK) method and further extend it to a relaxed variant (RGRBK) and a deterministic counterpart (MWRBK). We establish the convergence properties of the proposed methods. Numerical tests verify the theoretical findings, and we apply the proposed methods to color image restoration problems.
Paper Structure (13 sections, 11 theorems, 68 equations, 6 figures, 12 tables, 7 algorithms)

This paper contains 13 sections, 11 theorems, 68 equations, 6 figures, 12 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Block Kaczmarz methods and their convergence
  4. The general BK method
  5. The BK method with $B$ of full column rank
  6. The BK method with $B$ of full row rank
  7. GRBK method and its relaxed and deterministic variants
  8. The GRBK method
  9. The RGRBK method
  10. The MWRBK method
  11. Numerical examples
  12. An application to color image restoration
  13. Conclusions

Key Result

Theorem 2.1

ref12 Let matrix equation eq1.1 with $A\in \mathbb{R}^{m\times p}$, $B\in \mathbb{R}^{q\times n}$ and $C\in \mathbb{R}^{m\times n}$ be consistent. If $0<\alpha< 2 \, \|B\|^{-2}$, then the sequence $\{X^k\}_{k=0}^{\infty}$, obtained by RBK from an initial matrix $X^0\in \mathbb{R}^{p\times q}$, in wh where

Figures (6)

  • Figure 4.1: Schematic graph of $\delta_{k,\,\theta}$ with respect to $\theta$, and relations between RGRBK, GRBK, and MWRBK methods.
  • Figure 5.1: Convergence curves of Algorithm \ref{['ME-BK']} showing RSE versus IT for the test matrices listed in Table \ref{['table1']}.
  • Figure 5.2: For Set 5 (top row) and Set 1 (bottom row): the spectral radius, the convergence (RSE versus IT) for different $\alpha$, and the plots of $\rho^k$ and RSE versus IT for Algorithms \ref{['ME-BK1']} (top row) and \ref{['alg.AX=C']} (bottom row). Note that the upper bound on $\rho$ is minimized for $\alpha=1$ in these cases, but the actual minimizer $\alpha$ might differ from this value. Only in this figure, IT means the number of sweeps (rather than row-action steps).
  • Figure 6.1: Test images (top row) and blurred images (bottom row).
  • Figure 6.2: Relation between PSNR and CPU for the restored face (left), bird (middle), and mandril (right) images using different methods.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Theorem 2.1
  • Lemma 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Proposition 3.7
  • Remark 4.1
  • Theorem 4.2
  • ...and 4 more