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Greedy randomized Bregman-Kaczmarz method for constrained nonlinear systems of equations

Aqin Xiao, Junfeng Yin

Abstract

A greedy randomized nonlinear Bregman-Kaczmarz method by sampling the working index with residual information is developed for the solution of the constrained nonlinear system of equations. Theoretical analyses prove the convergence of the greedy randomized nonlinear Bregman-Kaczmarz method and its relaxed version. Numerical experiments verify the effectiveness of the proposed method,which converges faster than the existing nonlinear Bregman-Kaczmarz methods.

Greedy randomized Bregman-Kaczmarz method for constrained nonlinear systems of equations

Abstract

A greedy randomized nonlinear Bregman-Kaczmarz method by sampling the working index with residual information is developed for the solution of the constrained nonlinear system of equations. Theoretical analyses prove the convergence of the greedy randomized nonlinear Bregman-Kaczmarz method and its relaxed version. Numerical experiments verify the effectiveness of the proposed method,which converges faster than the existing nonlinear Bregman-Kaczmarz methods.
Paper Structure (9 sections, 7 theorems, 34 equations, 4 figures, 2 algorithms)

This paper contains 9 sections, 7 theorems, 34 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The greedy randomized nonlinear Bregman-Kaczmarz method
  4. Convergence analysis
  5. Numerical experiments
  6. Sparse solutions of quadratic equations
  7. Linear systems on the probability simplex
  8. Left stochastic decomposition
  9. Conclusions

Key Result

Proposition 2.1

Let $\alpha\in\mathbb{R}^d\setminus\{0\}$ and $\beta\in\mathbb{R}$ such that Then, for all $x\in\text{dom }\partial\varphi$ and $x^*\in\partial\varphi(x)$, the Bregman projection $\Pi_{\varphi, H(\alpha,\beta)}^{x^*}(x)$ exists and is unique, which is given by where $x_+^* = x^* - \hat{t}\alpha \in \partial\varphi(x_+)$ and $\hat{t}$ is a solution to

Figures (4)

  • Figure 1: Convergence curves of residual versus iterations for Problem \ref{['problem_LSQE']} with $(n,d)=(500, 100),s=10, \lambda=5$.
  • Figure 2: Convergence curves of residual versus iterations for Problem \ref{['problem_LSPS']} with Left: $A\sim\mathcal{N}(0,1)^{400\times300}$ and Right: $A\sim\mathcal{N}(0,1)^{300\times400}$.
  • Figure 3: Convergence curves of residual versus iterations for Problem \ref{['problem_LSPS']} with Left: $A\sim\mathcal{U}([0,1])^{300\times400}$ and Right: $A\sim\mathcal{U}([0.9,1])^{300\times400}$.
  • Figure 4: Convergence curves of residual versus iterations for Problem \ref{['problem_LSD']} with Left: $r=100, m=90$ and Right: $r=90, m=100$.

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.1
  • Theorem 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 5 more