SIAM Journal on Optimization, Volume 31, Issue 2, Page 1576-1603, January 2021.
Recently, in a series of papers [Y. Rahimi, C. Wang, H. Dong, and Y. Lou, SIAM J. Sci. Comput., 41 (2019), pp. A3649--A3672; C. Wang, M. Tao, J. Nagy, and Y. Lou, SIAM J. Imaging Sci., 14 (2021), pp. 749--777; C. Wang, M. Yan, and Y. Lou, IEEE Trans. Signal Process., 68 (2020), pp. 2660--2669; P. Yin, E. Esser, and J. Xin, Commun. Inf. Syst., 14 (2014), pp. 87--109], the ratio of $\ell_1$ and $\ell_2$ norms was proposed as a sparsity inducing function for noiseless compressed sensing. In this paper, we further study properties of such model in the noiseless setting, and propose an algorithm for minimizing $\ell_1$/$\ell_2$ subject to noise in the measurements. Specifically, we show that the extended objective function (the sum of the objective and the indicator function of the constraint set) of the model in [Y. Rahimi, C. Wang, H. Dong, and Y. Lou, SIAM J. Sci. Comput., 41 (2019), pp. A3649--A3672] satisfies the Kurdyka--Łojasiewicz (KL) property with exponent 1/2; this allows us to establish linear convergence of the algorithm proposed in [C. Wang, M. Yan, and Y. Lou, IEEE Trans. Signal Process., 68 (2020), pp. 2660--2669] (see equation 11) under mild assumptions. We next extend the $\ell_1$/$\ell_2$ model to handle compressed sensing problems with noise. We establish the solution existence for some of these models under the spherical section property [S. A. Vavasis, Derivation of Compressive Sensing Theorems from the Spherical Section Property, University of Waterloo, 2009; Y. Zhang, J. Oper. Res. Soc. China, 1 (2013), pp. 79--105] and extend the algorithm in [C. Wang, M. Yan, and Y. Lou, IEEE Trans. Signal Process., 68 (2020), pp. 2660--2669] (see equation 11) by incorporating moving-balls-approximation techniques [A. Auslender, R. Shefi, and M. Teboulle, SIAM J. Optim., 20 (2010), pp. 3232--3259] for solving these problems. We prove the subsequential convergence of our algorithm under mild conditions and establish global convergence of the whole sequence generated by our algorithm by imposing additional KL and differentiability assumptions on a specially constructed potential function. Finally, we perform numerical experiments on robust compressed sensing and basis pursuit denoising with residual error measured by $ \ell_2 $ norm or Lorentzian norm via solving the corresponding $\ell_1$/$\ell_2$ models by our algorithm. Our numerical simulations show that our algorithm is able to recover the original sparse vectors with reasonable accuracy.

中文翻译：

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基于L1/L2最小化的一些压缩感知模型分析与算法

SIAM 优化杂志，第 31 卷，第 2 期，第 1576-1603 页，2021 年 1 月。
最近，在一系列论文中[Y. Rahimi、C. Wang、H. Dong 和 Y. Lou、SIAM J. Sci。计算机，41 (2019)，第 A3649--A3672；C. Wang、M. Tao、J. Nagy 和 Y. Lou，SIAM J. Imaging Sci.，14 (2021)，第 749--777 页；C. Wang、M. Yan 和 Y. Lou，IEEE Trans。Signal Process., 68 (2020), pp. 2660--2669；P. Yin、E. Esser 和 J. Xin，Commun。信息。Syst., 14 (2014), pp. 87--109]，$\ell_1$ 和 $\ell_2$ 范数的比率被提出作为无噪声压缩感知的稀疏诱导函数。在本文中，我们进一步研究了这种模型在无噪声环境中的特性，并提出了一种在测量中最小化受噪声影响的$\ell_1$/$\ell_2$ 的算法。具体来说，我们在[Y. Rahimi、C. Wang、H. Dong 和 Y. 娄，暹罗 J. Sci。Comput., 41 (2019), pp. A3649--A3672] 满足 Kurdyka--Łojasiewicz (KL) 属性，指数为 1/2；这使我们能够建立 [C. Wang、M. Yan 和 Y. Lou，IEEE Trans。Signal Process., 68 (2020), pp. 2660--2669]（见方程 11）在温和的假设下。我们接下来扩展 $\ell_1$/$\ell_2$ 模型来处理带有噪声的压缩感知问题。我们在球截面属性下建立了其中一些模型的解存在性 [SA Vavasis, Derivation of Compressive Sensing Theorems from the Spherical Section Property, University of Waterloo, 2009; Y.张，J.歌剧。水库。社会党。China, 1 (2013), pp. 79--105] 并在 [C. Wang、M. Yan 和 Y. Lou，IEEE Trans。信号处理。，68（2020），第。2660--2669]（参见方程 11）通过结合移动球近似技术 [A. Auslender, R. Shefi 和 M. Teboulle, SIAM J. Optim., 20 (2010), pp. 3232--3259] 用于解决这些问题。我们证明了我们的算法在温和条件下的后续收敛性，并通过对特殊构造的势函数施加额外的 KL 和可微性假设来建立由我们的算法生成的整个序列的全局收敛性。最后，我们通过我们的算法求解相应的 $\ell_1$/$\ell_2$ 模型，对鲁棒压缩感知和基追踪去噪进行数值实验，残差由 $\ell_2$ 范数或洛伦兹范数测量。我们的数值模拟表明我们的算法能够以合理的精度恢复原始稀疏向量。