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Reverse Quantile-RK and its Application to Quantile-RK

Emeric Battaglia, Anna Ma

TL;DR

It is shown that the proposed acceleration converges faster than the randomized Kaczmarz algorithm, and can be used in conjunction with the quantile‐randomized Kaczmarz algorithm, without adding additional computational complexity, to produce both a fast and robust iterative method for solving large, sparsely corrupted linear systems that are sufficiently well‐conditioned.

Abstract

When solving linear systems $Ax=b$, $A$ and $b$ are given, but the measurements $b$ often contain corruptions. Inspired by recent work on the quantile-randomized Kaczmarz method, we propose an acceleration of the randomized Kaczmarz method using quantile information. We show that the proposed acceleration converges faster than the randomized Kaczmarz algorithm. In addition, we show that our proposed approach can be used in conjunction with the quantile-randomized Kaczamrz algorithm, without adding additional computational complexity, to produce both a fast and robust iterative method for solving large, sparsely corrupted linear systems. Our extensive experimental results support the use of the revised algorithm.

Reverse Quantile-RK and its Application to Quantile-RK

TL;DR

It is shown that the proposed acceleration converges faster than the randomized Kaczmarz algorithm, and can be used in conjunction with the quantile‐randomized Kaczmarz algorithm, without adding additional computational complexity, to produce both a fast and robust iterative method for solving large, sparsely corrupted linear systems that are sufficiently well‐conditioned.

Abstract

When solving linear systems , and are given, but the measurements often contain corruptions. Inspired by recent work on the quantile-randomized Kaczmarz method, we propose an acceleration of the randomized Kaczmarz method using quantile information. We show that the proposed acceleration converges faster than the randomized Kaczmarz algorithm. In addition, we show that our proposed approach can be used in conjunction with the quantile-randomized Kaczamrz algorithm, without adding additional computational complexity, to produce both a fast and robust iterative method for solving large, sparsely corrupted linear systems. Our extensive experimental results support the use of the revised algorithm.

Paper Structure

This paper contains 14 sections, 10 theorems, 76 equations, 4 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Background
  4. Contributions
  5. Analysis of the Reverse-Quantile Random Kaczmarz Method
  6. Useful Results
  7. The Main Result
  8. Analysis of the Double-Quantile Random Kaczmarz Method
  9. Prerequisites
  10. The Bound on dqRK
  11. Numerical Results
  12. Numerical Results for rqRK
  13. Numerical Results for dqRK
  14. Final Remarks

Key Result

Theorem 1

Assume $A$ is an over-determined, full-rank, and ${x^\ast}$ satisfies $A{x^\ast}=b$. Suppose $x_0$ satisfies $\langle x_0,a_i\rangle=b_i$ for some $i\in[m]$. Then rqRK with parameter $\frac{1}{m}\leqslant q \leqslant \frac{m-1}{m}$ yields In particular, for row-normalized matrices, this simplifies to

Figures (4)

  • Figure 1: Comparison between sample runs of RK and rqRK for varying lower quantiles $q$.
  • Figure 2: Comparison between sample runs of qRK and dqRK for $q_0=0.6$, $q_1=0.8$ on a linear system with sparse noise \ref{['eq:corruptsyseq']} with $\beta = 0.05$.
  • Figure 3: Comparison between sample runs of dqRK, using varying quantile parameters. The error proportion parameter $\beta$ is fixed at $0.05$.
  • Figure 4: Comparison between a sample run of RK, qRK, and dqRK for $q_0=0.6$, $q_1=0.8$, and $\beta = 0.05$ on the row-normalized ash958 matrix from SparseSuite sparsesuite.

Theorems & Definitions (15)

  • Theorem 1: Main Result
  • Theorem 2
  • Theorem 3: qRK, Main Theorem
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Theorem 3: Main Result
  • proof
  • Corollary 2
  • ...and 5 more