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A projected gradient method for nonlinear inverse problems with 𝛼ℓ1 − 𝛽ℓ2 sparsity regularization

  • Zhuguang Zhao and Liang Ding EMAIL logo
Published/Copyright: July 25, 2023
Journal of Inverse and Ill-posed Problems
From the journal Journal of Inverse and Ill-posed Problems Volume 32 Issue 3

Abstract

The non-convex α ⁢ ∥ ⋅ ∥ ℓ 1 − β ⁢ ∥ ⋅ ∥ ℓ 2 ( α ≥ β ≥ 0 ) regularization is a new approach for sparse recovery. A minimizer of the α ⁢ ∥ ⋅ ∥ ℓ 1 − β ⁢ ∥ ⋅ ∥ ℓ 2 regularized function can be computed by applying the ST-( α ⁢ ℓ 1 − β ⁢ ℓ 2 ) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). Unfortunately, It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. Nevertheless, the current applicability of the PG method is limited to linear inverse problems. In this paper, we extend the PG method based on a surrogate function approach to nonlinear inverse problems with the α ⁢ ∥ ⋅ ∥ ℓ 1 − β ⁢ ∥ ⋅ ∥ ℓ 2 ( α ≥ β ≥ 0 ) regularization in the finite-dimensional space R n . It is shown that the presented algorithm converges subsequentially to a stationary point of a constrained Tikhonov-type functional for sparsity regularization. Numerical experiments are given in the context of a nonlinear compressive sensing problem to illustrate the efficiency of the proposed approach.

Keywords: Projected gradient method; nonlinear inverse problems; surrogate function approach
MSC 2010: 49M37; 65K05

Funding source: Fundamental Research Funds for the Central Universities

Award Identifier / Grant number: 2572021DJ03

Funding source: Postdoctoral Scientific Research Development Fund of Heilongjiang Province

Award Identifier / Grant number: LBH-Q16008

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 41304093

Funding statement: The work of the second author was supported by the Fundamental Research Funds for the Central Universities (no. 2572021DJ03), Heilongjiang Postdoctoral Research Developmental Fund (no. LBH-Q16008) and the National Natural Science Foundation of China (no. 41304093).

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Received: 2023-01-27
Revised: 2023-05-14
Accepted: 2023-06-14
Published Online: 2023-07-25
Published in Print: 2024-06-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Stability properties for a class of inverse problems
  3. Inverse scattering problem for nonstrict hyperbolic system on the half-axis with a nonzero boundary condition
  4. Identification of the time-dependent source term in a Kuramoto–Sivashinsky equation
  5. Direct numerical algorithm for calculating the heat flux at an inaccessible boundary
  6. The game model with multi-task for image denoising and edge extraction
  7. On the X-ray transform of planar symmetric tensors
  8. Inverse vector problem of diffraction by inhomogeneous body with a piecewise smooth permittivity
  9. Acquiring elastic properties of thin composite structure from vibrational testing data
  10. Inverse nodal problem for diffusion operator on a star graph with nonhomogeneous edges
  11. Convergence analysis of Inexact Newton–Landweber iteration with frozen derivative in Banach spaces
  12. A projected gradient method for nonlinear inverse problems with 𝛼ℓ1 − 𝛽ℓ2 sparsity regularization
  13. Correctness and regularization of stochastic problems
  14. A layer potential approach to inverse problems in brain imaging
  15. Inverse problem for Dirac operators with two constant delays
https://doi.org/10.1515/jiip-2023-0010
Keywords for this article
Projected gradient method; nonlinear inverse problems; surrogate function approach

Articles in the same Issue

  1. Frontmatter
  2. Stability properties for a class of inverse problems
  3. Inverse scattering problem for nonstrict hyperbolic system on the half-axis with a nonzero boundary condition
  4. Identification of the time-dependent source term in a Kuramoto–Sivashinsky equation
  5. Direct numerical algorithm for calculating the heat flux at an inaccessible boundary
  6. The game model with multi-task for image denoising and edge extraction
  7. On the X-ray transform of planar symmetric tensors
  8. Inverse vector problem of diffraction by inhomogeneous body with a piecewise smooth permittivity
  9. Acquiring elastic properties of thin composite structure from vibrational testing data
  10. Inverse nodal problem for diffusion operator on a star graph with nonhomogeneous edges
  11. Convergence analysis of Inexact Newton–Landweber iteration with frozen derivative in Banach spaces
  12. A projected gradient method for nonlinear inverse problems with 𝛼ℓ1 − 𝛽ℓ2 sparsity regularization
  13. Correctness and regularization of stochastic problems
  14. A layer potential approach to inverse problems in brain imaging
  15. Inverse problem for Dirac operators with two constant delays
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