In this paper, we propose a new method for support detection and estimation of sparse and approximately sparse signals from compressed measurements. Using a double Laplace mixture model as the parametric representation of the signal coefficients, the problem is formulated as a weighted ℓ 1 minimization. Then, we introduce a new family of iterative shrinkage-thresholding algorithms based on double Laplace mixture models. They preserve the computational simplicity of classical ones and improve iterative estimation by incorporating soft support detection. In particular, at each iteration, by learning the components that are likely to be nonzero from the current MAP signal estimate, the shrinkage-thresholding step is adaptively tuned and optimized. Unlike other adaptive methods, we are able to prove, under suitable conditions, the convergence of the proposed methods to a local minimum of the weighted ℓ 1 minimization. Moreover, we also provide an upper bound on the reconstruction error. Finally, we show through numerical experiments that the proposed methods outperform classical shrinkage-thresholding in terms of rate of convergence, accuracy, and of sparsity-undersampling trade-off.

中文翻译：

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通过拉普拉斯混合模型改进了稀疏信号恢复的迭代收缩阈值。

在本文中，我们提出了一种新方法，用于支持从压缩测量中检测和估计稀疏和近似稀疏信号。使用双拉普拉斯混合模型作为信号系数的参数表示，该问题被表述为加权 ℓ 1 最小化。然后，我们引入了基于双拉普拉斯混合模型的新的迭代收缩阈值算法系列。它们保留了经典计算的简单性，并通过结合软支撑检测来改进迭代估计。特别是，在每次迭代时，通过学习当前 MAP 信号估计中可能非零的分量，自适应地调整和优化收缩阈值步骤。与其他自适应方法不同，我们能够在适当的条件下证明所提出的方法收敛到加权 ℓ 1 最小化的局部最小值。此外，我们还提供了重建误差的上限。最后，我们通过数值实验表明，所提出的方法在收敛速度、准确性和稀疏性欠采样权衡方面优于经典的收缩阈值。