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Hard Thresholding Pursuit Algorithms for Least Absolute Deviations Problem

Jiao Xu, Peng Li, Bing Zheng

TL;DR

The robustness of the variant of adaptive iterative hard thresholding to outliers, known as graded fast hard thresholding pursuit (GFHTP) algorithm, is explored, which consistently outperforms competing algorithms in terms of both robustness and computational efficiency.

Abstract

Least absolute deviations (LAD) is a statistical optimality criterion widely utilized in scenarios where a minority of measurements are contaminated by outliers of arbitrary magnitudes. In this paper, we delve into the robustness of the variant of adaptive iterative hard thresholding to outliers, known as graded fast hard thresholding pursuit (GFHTP$_1$) algorithm. Unlike the majority of the state-of-the-art algorithms in this field, GFHTP$_1$ does not require prior information about the signal's sparsity. Moreover, its design is parameterless, which not only simplifies the implementation process but also removes the intricacies of parameter optimization. Numerical experiments reveal that the GFHTP$_1$ algorithm consistently outperforms competing algorithms in terms of both robustness and computational efficiency.

Hard Thresholding Pursuit Algorithms for Least Absolute Deviations Problem

TL;DR

The robustness of the variant of adaptive iterative hard thresholding to outliers, known as graded fast hard thresholding pursuit (GFHTP) algorithm, is explored, which consistently outperforms competing algorithms in terms of both robustness and computational efficiency.

Abstract

Least absolute deviations (LAD) is a statistical optimality criterion widely utilized in scenarios where a minority of measurements are contaminated by outliers of arbitrary magnitudes. In this paper, we delve into the robustness of the variant of adaptive iterative hard thresholding to outliers, known as graded fast hard thresholding pursuit (GFHTP) algorithm. Unlike the majority of the state-of-the-art algorithms in this field, GFHTP does not require prior information about the signal's sparsity. Moreover, its design is parameterless, which not only simplifies the implementation process but also removes the intricacies of parameter optimization. Numerical experiments reveal that the GFHTP algorithm consistently outperforms competing algorithms in terms of both robustness and computational efficiency.
Paper Structure (35 sections, 12 theorems, 60 equations, 11 figures, 3 tables, 2 algorithms)

This paper contains 35 sections, 12 theorems, 60 equations, 11 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Related Work
  4. Least Absolute Deviations
  5. Hard Thresholding Algorithms
  6. Motivation and Contributions
  7. Organization and Notations
  8. Algorithms and Theoretical Results
  9. Our Algorithms
  10. Parameter Description
  11. Fast Hard Thresholding Pursuit (FHTP$_1$) Algorithm
  12. Graded Fast Hard Thresholding Pursuit (GFHTP$_{1}$) Algorithm
  13. Theoretical Results
  14. Foundational Assumption: Restricted 1-Isometry Property (RIP$_1$)
  15. Error Estimation for General Sparse Signals
  16. ...and 20 more sections

Key Result

Theorem 2.2

Given an $s$-sparse vector $\mathbf{x}_0\in \mathbb{R}^n$ and the measurements $\mathbf{b}= \mathbf{Ax}_0+\boldsymbol{\eta}$, where $\boldsymbol{\eta}$ is a vector of outliers with support $T$ and cardinality $|T|= pm$. Assume that the matrix $\mathbf{A}$ satisfies RIP$_1$ with RIC$_1$ satisfying $0 where $\mu_{k,l}$ satisfies the following inequality Here $c_k= (2-2p)(1-\delta_{2k+s-1})-(1+\delt

Figures (11)

  • Figure 1: The maximum value of the outliers proportion $p$ under general case.
  • Figure 2: The maximum value of the outliers proportion $p$ under 'flat' vector.
  • Figure 3: Relative error, average CPU time, the sparsity of the recovered signal for the GFHTP$_1$ algorithm using Gaussian outliers.
  • Figure 4: Numerical performance for the GFHTP$_1$ algorithm using Gaussian outliers when various $L$.
  • Figure 5: Numerical performance for the GFHTP$_1$ algorithm when various $\tau$.
  • ...and 6 more figures

Theorems & Definitions (23)

  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 13 more