Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A fast block nonlinear Bregman-Kaczmarz method with averaging for nonlinear sparse signal recovery

Aqin Xiao, Xiangyu Gao, Jun-Feng Yin

TL;DR

The paper tackles sparse recovery from nonlinear measurements by addressing the nonlinear system $F(x)=0$ with a sparsity-promoting objective $\varphi$ and nonlinear Bregman projections. It introduces the averaging block nonlinear Bregman-Kaczmarz (ABNBK) method, offering constant and adaptive step-size variants that leverage block averaging and Bregman distances $D_\varphi^{x^*}$. The authors establish convergence rates under the local tangential cone condition and provide explicit linear-rate bounds for both constant and adaptive step-sizes, linked to problem conditioning. Numerical experiments demonstrate that ABNBK variants achieve faster convergence (fewer iterations and lower CPU time) than existing NBK and MRNBK methods across Gaussian and DCT-type measurement scenarios, confirming scalability to large nonlinear problems. Overall, the approach delivers provable efficiency gains and practical robustness for nonlinear sparse signal recovery.

Abstract

Recovery of a sparse signal from a nonlinear system arises in many practical applications including compressive sensing, image reconstruction and machine learning. In this paper, a fast block nonlinear Bregman-Kaczmarz method with averaging is presented for nonlinear sparse signal recovery problems. Theoretical analysis proves that the averaging block nonlinear Bregman-Kaczmarz method with both constant stepsizes and adaptive stepsizes are convergent. Numerical experiments demonstrate the effectiveness of the averaging block nonlinear Bregman-Kaczmarz method, which converges faster than the existing nonlinear Bregman-Kaczmarz methods.

A fast block nonlinear Bregman-Kaczmarz method with averaging for nonlinear sparse signal recovery

TL;DR

The paper tackles sparse recovery from nonlinear measurements by addressing the nonlinear system with a sparsity-promoting objective and nonlinear Bregman projections. It introduces the averaging block nonlinear Bregman-Kaczmarz (ABNBK) method, offering constant and adaptive step-size variants that leverage block averaging and Bregman distances . The authors establish convergence rates under the local tangential cone condition and provide explicit linear-rate bounds for both constant and adaptive step-sizes, linked to problem conditioning. Numerical experiments demonstrate that ABNBK variants achieve faster convergence (fewer iterations and lower CPU time) than existing NBK and MRNBK methods across Gaussian and DCT-type measurement scenarios, confirming scalability to large nonlinear problems. Overall, the approach delivers provable efficiency gains and practical robustness for nonlinear sparse signal recovery.

Abstract

Recovery of a sparse signal from a nonlinear system arises in many practical applications including compressive sensing, image reconstruction and machine learning. In this paper, a fast block nonlinear Bregman-Kaczmarz method with averaging is presented for nonlinear sparse signal recovery problems. Theoretical analysis proves that the averaging block nonlinear Bregman-Kaczmarz method with both constant stepsizes and adaptive stepsizes are convergent. Numerical experiments demonstrate the effectiveness of the averaging block nonlinear Bregman-Kaczmarz method, which converges faster than the existing nonlinear Bregman-Kaczmarz methods.

Paper Structure

This paper contains 7 sections, 9 theorems, 41 equations, 8 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Fast block nonlinear Bregman-Kaczmarz method
  3. Convergence analysis
  4. Averaging block nonlinear Bregman-Kaczmarz methods with constant stepsize
  5. Averaging block nonlinear Bregman-Kaczmarz methods with adaptive stepsize
  6. Numerical experiments
  7. Conclusions

Key Result

Theorem 2.1

Let 2024GLW hold true, $\varphi$ be $\sigma$-strongly convex and $M$-smooth. Let each function $F_i$ satisfies the local tangential cone condition with some $\eta\in (0,\frac{\sigma}{2M})$, $\hat{x}\in S$ and $x_0\in B_{r,\varphi}(\hat{x})$. Moreover, assume that the Jacobian $F'(x)$ has full column where $\kappa_{\min}= \min\limits_{x\in B_{r,\varphi}(\hat{x})} \min\limits_{\|y\|_2=1} \frac{\|F'(

Figures (8)

  • Figure 1: Convergence curves for Example \ref{['ex1:GAUmatrix']} with $sp=0.1$.
  • Figure 2: Convergence curves for Example \ref{['ex1:GAUmatrix']} with $sp=0.05$.
  • Figure 3: The exact signal and recovered signals for Example \ref{['ex1:GAUmatrix']} with $sp=0.1$.
  • Figure 4: The exact signal and recovered signals for Example \ref{['ex1:GAUmatrix']} with $sp=0.05$.
  • Figure 5: Convergence curves for Example \ref{['ex1:DCTmatrix']} with $sp=0.1$.
  • ...and 3 more figures

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 8 more