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Newton-Direction-Based ReLU-Thresholding Methods for Nonnegative Sparse Signal Recovery

Ning Bian, Zhong-Feng Sun, Yun-Bin Zhao, Jin-Chuan Zhou, Nan Meng

TL;DR

Newton-Direction-Based ReLU-Thresholding (NDRT) and its enhanced variant, Newton-Direction-Based ReLU-Thresholding Pursuit (NDRTP) are proposed that can guarantee exact recovery of nonnegative sparse signals when the measurement matrix satisfies a certain condition.

Abstract

Nonnegative sparse signal recovery has been extensively studied due to its broad applications. Recent work has integrated rectified linear unit (ReLU) techniques to enhance existing recovery algorithms. We merge Newton-type thresholding with ReLU-based approaches to propose two algorithms: Newton-Direction-Based ReLU-Thresholding (NDRT) and its enhanced variant, Newton-Direction-Based ReLU-Thresholding Pursuit (NDRTP). Theoretical analysis iindicates that both algorithms can guarantee exact recovery of nonnegative sparse signals when the measurement matrix satisfies a certain condition.. Numerical experiments demonstrate NDRTP achieves competitive performance compared to several existing methods in both noisy and noiseless scenarios.

Newton-Direction-Based ReLU-Thresholding Methods for Nonnegative Sparse Signal Recovery

TL;DR

Newton-Direction-Based ReLU-Thresholding (NDRT) and its enhanced variant, Newton-Direction-Based ReLU-Thresholding Pursuit (NDRTP) are proposed that can guarantee exact recovery of nonnegative sparse signals when the measurement matrix satisfies a certain condition.

Abstract

Nonnegative sparse signal recovery has been extensively studied due to its broad applications. Recent work has integrated rectified linear unit (ReLU) techniques to enhance existing recovery algorithms. We merge Newton-type thresholding with ReLU-based approaches to propose two algorithms: Newton-Direction-Based ReLU-Thresholding (NDRT) and its enhanced variant, Newton-Direction-Based ReLU-Thresholding Pursuit (NDRTP). Theoretical analysis iindicates that both algorithms can guarantee exact recovery of nonnegative sparse signals when the measurement matrix satisfies a certain condition.. Numerical experiments demonstrate NDRTP achieves competitive performance compared to several existing methods in both noisy and noiseless scenarios.
Paper Structure (10 sections, 6 theorems, 76 equations, 3 figures, 1 table, 3 algorithms)

This paper contains 10 sections, 6 theorems, 76 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Algorithms
  3. Notation and basic inequalities
  4. Algorithms
  5. Analysis of Algorithms
  6. Numerical Experiments
  7. Gradient Projection Algorithm for Nonnegative Least Squares Subproblems
  8. Parameter configurations of NDRT and NDRTP
  9. Comparison to some existing algorithms
  10. Conclusion

Key Result

Lemma 2.2

Let ${\bm A} \in \mathbb{R}^{m \times n}$ with $m \ll n$ be a matrix. Let ${\bm u} \in \mathbb{R}^m$, ${\bm v}, {\bm w} \in \mathbb{R}^n$, and let $k$ be a positive integer. Then the following inequalities hold:

Figures (3)

  • Figure 1: Success frequencies of NDRTP with varying parameters in noiseless settings
  • Figure 2: Comparison of success frequencies and runtime of algorithms in noiseless settings.
  • Figure 3: Comparison of success frequencies and runtime of algorithms in noisy settings.

Theorems & Definitions (12)

  • Definition 2.1: See CT
  • Lemma 2.2: See SF
  • Lemma 2.3: See ZH
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • ...and 2 more