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<p>Recently, the nonconvex fraction function has been well studied to recover the sparse signals, and the numerical results have shown that the fraction function performs very well in sparse signal recovery. However, the proposed FP algorithm for solving the fraction function minimization can only be proven to converge to a stationary point owing to the nonconvexity of the fraction function, and it is unclear what this stationary point is. More specifically, due to the nonconvex nature, this stationary point is hardly possible to be regarded as the global minimizer. In this paper, different from the previous proposed FP algorithm, a new algorithm and its adaptive version algorithm are studied to solve the fraction function minimization again. Under some conditions, both these two new algorithms can converge to the neighborhood of the global optimal solution of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\ell _{0}\)</EquationSource> </InlineEquation>-norm minimization problem. The established convergence results of the new algorithms can provide a theoretical guarantee for a wide range of applications of the nonconvex fraction function in compressed sensing relevant problems. We also provide some numerical simulations to verify the performance of the proposed adaptive algorithm, and the numerical results show the effectiveness of the adaptive algorithm in sparse signal recovery.</p>

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Truncated iterative thresholding algorithm for compressed sensing via fraction function

  • 期刊论文

  • 发表日期：

    2026年1月21日

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  • Angang Cui,
  • Lijun Zhang,
  • Shengli Xue,
  • Hong Yang

摘要

Recently, the nonconvex fraction function has been well studied to recover the sparse signals, and the numerical results have shown that the fraction function performs very well in sparse signal recovery. However, the proposed FP algorithm for solving the fraction function minimization can only be proven to converge to a stationary point owing to the nonconvexity of the fraction function, and it is unclear what this stationary point is. More specifically, due to the nonconvex nature, this stationary point is hardly possible to be regarded as the global minimizer. In this paper, different from the previous proposed FP algorithm, a new algorithm and its adaptive version algorithm are studied to solve the fraction function minimization again. Under some conditions, both these two new algorithms can converge to the neighborhood of the global optimal solution of \(\ell _{0}\) -norm minimization problem. The established convergence results of the new algorithms can provide a theoretical guarantee for a wide range of applications of the nonconvex fraction function in compressed sensing relevant problems. We also provide some numerical simulations to verify the performance of the proposed adaptive algorithm, and the numerical results show the effectiveness of the adaptive algorithm in sparse signal recovery.