The non-convex α ⁢ ∥ ⋅ ∥ ℓ 1 − β ⁢ ∥ ⋅ ∥ ℓ 2 \alpha\lVert\,{\cdot}\,\rVert_{\ell_{1}}-\beta\lVert\,{\cdot}\,\rVert_{\ell_{2}} ( α ≥ β ≥ 0 \alpha\geq\beta\geq 0 ) regularization is a new approach for sparse recovery. A minimizer of the α ⁢ ∥ ⋅ ∥ ℓ 1 − β ⁢ ∥ ⋅ ∥ ℓ 2 \alpha\lVert\,{\cdot}\,\rVert_{\ell_{1}}-\beta\lVert\,{\cdot}\,\rVert_{\ell_{2}} regularized function can be computed by applying the ST-( α ⁢ ℓ 1 − β ⁢ ℓ 2 \alpha\ell_{1}-\beta\ell_{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 \alpha\lVert\,{\cdot}\,\rVert_{\ell_{1}}-\beta\lVert\,{\cdot}\,\rVert_{\ell_{2}} ( α ≥ β ≥ 0 \alpha\geq\beta\geq 0 ) regularization in the finite-dimensional space R n \mathbb{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.

中文翻译：

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具有 𝛼ℓ1 − 𝛽ℓ2 稀疏正则化的非线性反问题的投影梯度法

非凸 α ⁢ ∥ ⋅ ∥ ℓ 1 - β ⁢ ∥ ⋅ ∥ ℓ 2 \alpha\lVert\,{\cdot}\,\rVert_{\ell_{1}}-\beta\lVert\,{\cdot}\,\rVert_{\ell_{2}} （ α ≥ β ≥ 0 \alpha\geq\beta\geq 0 ）正则化是稀疏恢复的一种新方法。的最小化 α ⁢ ∥ ⋅ ∥ ℓ 1 - β ⁢ ∥ ⋅ ∥ ℓ 2 \alpha\lVert\,{\cdot}\,\rVert_{\ell_{1}}-\beta\lVert\,{\cdot}\,\rVert_{\ell_{2}} 正则化函数可以通过应用 ST-( α ⁢ ℓ 1 - β ⁢ ℓ 2 \alpha\ell_{1}-\beta\ell_{2} ）算法类似于经典的迭代软阈值算法（ISTA）。不幸的是，众所周知，ISTA 收敛速度相当慢，而 ISTA 的更快替代方法是投影梯度（PG）方法。然而，PG方法目前的适用性仅限于线性反问题。在本文中，我们将基于代理函数方法的 PG 方法扩展到非线性反问题，其中 α ⁢ ∥ ⋅ ∥ ℓ 1 - β ⁢ ∥ ⋅ ∥ ℓ 2 \alpha\lVert\,{\cdot}\,\rVert_{\ell_{1}}-\beta\lVert\,{\cdot}\,\rVert_{\ell_{2}} （ α ≥ β ≥ 0 \alpha\geq\beta\geq 0 ) 有限维空间中的正则化 右 n \mathbb{R}^{n} 。结果表明，所提出的算法随后收敛到稀疏正则化约束 Tikhonov 型函数的驻点。在非线性压缩感知问题的背景下给出了数值实验，以说明所提出方法的效率。