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Relaxed Greedy Randomized Kaczmarz with Signal Averaging for Solving Doubly-Noisy Linear Systems

Lu Zhang, Jinchuan Zeng, Hui Zhang

Abstract

Large-scale linear systems of the form $Ax=b$ are often doubly-noisy, in the sense that both its measurement matrix $A$ and measurement vector $b$ are noisy. In this paper, we extend the relaxed greedy randomized Kaczmarz (RGRK) method to the doubly-noisy systems to accelerate convergence. However, RGRK fails to converge to the least-squares solution for doubly-noisy systems. To address this limitation, we propose a simple modification: averaging multiple measurements instead of using a single measurement. The proposed RGRK with signal averaging (RGRK-SA) converges to the solution of doubly-noisy systems at a polynomial rate. Numerical experiments demonstrate that both RGRK and RGRK-SA outperform the classical randomized Kaczmarz method, and RGRK-SA has a higher accuracy.

Relaxed Greedy Randomized Kaczmarz with Signal Averaging for Solving Doubly-Noisy Linear Systems

Abstract

Large-scale linear systems of the form are often doubly-noisy, in the sense that both its measurement matrix and measurement vector are noisy. In this paper, we extend the relaxed greedy randomized Kaczmarz (RGRK) method to the doubly-noisy systems to accelerate convergence. However, RGRK fails to converge to the least-squares solution for doubly-noisy systems. To address this limitation, we propose a simple modification: averaging multiple measurements instead of using a single measurement. The proposed RGRK with signal averaging (RGRK-SA) converges to the solution of doubly-noisy systems at a polynomial rate. Numerical experiments demonstrate that both RGRK and RGRK-SA outperform the classical randomized Kaczmarz method, and RGRK-SA has a higher accuracy.

Paper Structure

This paper contains 16 sections, 5 theorems, 52 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization
  4. The related work
  5. The sampling methods
  6. The denoising techniques
  7. Notation
  8. The RGRK method for doubly-noisy linear systems
  9. Multiplicative Noise
  10. The RGRK method with signal averaging
  11. Numerical Experiments
  12. The effect of $\theta$ and $N$
  13. Comparisons with RK
  14. The simulated data
  15. The real-world data
  16. ...and 1 more sections

Key Result

Proposition 3.1

Let $\{w_k\}_{k\in\mathbb{N}}$ be the sequence generated by Algorithm al1. Then for $k\in\mathbb{N}^{+}$ we have where $\gamma=\max_{1\leq i\leq m} \sum_{j=1,j\neq i}^m \|\tilde{a}_{i}\|^2$; and for $k=0$ it holds that

Figures (4)

  • Figure 1: The effect of $\theta$ on RGRK and RGRK-SA
  • Figure 2: The effect of $N$ on RGRK-SA
  • Figure 3: Relative error and computing time after 4000 iterations of RK, RGRK, and RGRK-SA for simulated linear systems
  • Figure 4: Relative error and computing time after 2000 iterations of RK, RGRK, and RGRK-SA for real-world linear systems

Theorems & Definitions (11)

  • Proposition 3.1: Theorem 2.1 in bai2018relaxed
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Lemma 3.1
  • Remark 4.1
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Remark 4.3
  • ...and 1 more