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Accelerating monotone fast iterative shrinkage–thresholding algorithm with sequential subspace optimization for sparse recovery

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Abstract

The present paper focuses on accelerating monotone fast iterative shrinkage–thresholding algorithm (MFISTA) that is popular to solve the basis pursuit denoising problem for sparse recovery. Inspired by a recent work that accelerates MFISTA with line search, we alternatively use a much more effective speed-up option, termed sequential subspace optimization. Furthermore, instead of manually setting the number of previous propagation directions in the subspace beforehand, we propose an adaptive method to set it. Additionally, for approximating the absolute value function, we analyze the superiority of a smooth version used in this paper over the one recommended in a previous work, and give an analytical closed-form expression for the shrinkage operator corresponding to the smooth approximation. The experiments presented here show that the proposed method achieves faster convergence speeds in terms of iteration and run-time.

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Notes

  1. It is applied to a vector or matrix element-wise.

  2. This is the Lipschitz constant with respect to the gradient of the quadratic term in (1).

  3. The objective function could be non-smooth since the gradient direction is not necessary.

  4. We use the termination criterion (12) as an example for presentation.

  5. When other optimization methods that do not require a smooth objective function are employed for line search, MFISTA-LS can directly solve (1).

  6. \(\mathbf {A}\) is a normal matrix if \(\mathbf {A}^\mathrm{T}\mathbf {A}=\mathbf {A}\mathbf {A}^\mathrm{T}\).

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  1. School of Electronic and Information Engineering, South China University of Technology, Guangzhou, 510640, China

    Tao Zhu

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Correspondence to Tao Zhu.

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Zhu, T. Accelerating monotone fast iterative shrinkage–thresholding algorithm with sequential subspace optimization for sparse recovery. SIViP 14, 771–780 (2020). https://doi.org/10.1007/s11760-019-01603-4

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