压缩感知中的分布鲁棒优化模型及其求解方法
Distributed Robust Optimization Model and Its Solution Method in Compressed Sensing
摘要: 压缩感知(CS)理论作为当前先采样后压缩方法的替代方法引起了很多的关注，相关学者已经提出很多CS恢复算法如CoSaMP等，而带噪声的压缩感知模型可以表示为l1-范数问题，由于不易判断矩阵的有限等距性(RIP)以及观测矩阵的不确定性，考虑将该l1-范数问题建模为分布鲁棒优化问题，再借助KL-散度、函数变换以及内部最大化期望函数等方法得到分布鲁棒优化问题的等价形式。
Abstract: Compressed sensing (CS) theory has attracted a lot of attention as an alternative to the current sampling-then-compression method. Relevant scholars have proposed many CS recovery algorithms such as CoSaMP, etc., and the noisy compressed sensing model can be expressed as an l1-norm problem. Since it is difficult to judge the finite isometric property (RIP) of the matrix and the uncertainty of the observation matrix, consider modeling the l1-norm problem as a distributed robust optimization problem, and then use KL-divergence, function transformation and internal maximization methods such as the expectation function obtain equivalent forms of distributed robust optimization problems.
文章引用：李达臣. 压缩感知中的分布鲁棒优化模型及其求解方法[J]. 应用数学进展, 2024, 13(3): 1036-1045. https://doi.org/10.12677/aam.2024.133098

参考文献

[1] Tropp, J.A. and Gilbert, A.C. (2007) Signal Recovery from Random Measurements via Orthogonal Matching Pursuit. IEEE Transactions on Information Theory, 53, 4655-4666. [Google Scholar] [CrossRef] 
[2] Wang, J., Kwon, S. and Shim, B. (2012) Generalized Orthogonal Matching Pursuit. IEEE Transactions on Signal Pro-cessing, 60, 6202-6216. [Google Scholar] [CrossRef] 
[3] Wright, S.J., Nowak, R.D. and Figueiredo, M.A.T. (2009) Sparse Reconstruction by Separable Approximation. IEEE Transactions on Signal Processing, 57, 2479-2493. [Google Scholar] [CrossRef] 
[4] Chen, S.S., Donoho, D.L. and Saunders, M.A. (2001) Atomic Decomposition by Basis Pursuit. SIAM Review, 43, 129-159. [Google Scholar] [CrossRef] 
[5] Candès, E.J., Romberg, J. and Tao, T. (2006) Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information. IEEE Transactions on Information Theory, 52, 489-509. [Google Scholar] [CrossRef] 
[6] Gui, G., Wan, Q., Peng, W., et al. (2010) Sparse Multipath Channel Estimation Using Compressive Sampling Matching Pursuit Algorithm. arXiv:1005.2270.
[7] Goldstein, T. and Osher, S. (2009) The Split Bregman Method for L1-Regularized Problems. SIAM Journal on Imaging Sciences, 2, 323-343. [Google Scholar] [CrossRef] 
[8] Zhang, J., Zhao, D., Zhao, C., et al. (2012) Image Compressive Sensing Recovery via Collaborative Sparsity. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 2, 380-391. [Google Scholar] [CrossRef] 
[9] Zhang, J., Zhao, D., Zhao, C., et al. (2012) Compressed Sensing Recovery via Collaborative Sparsity. 2012 Data Compression Conference, Snowbird, 10-12 April 2012, 287-296. [Google Scholar] [CrossRef] 
[10] Donoho, D.L. (2006) Compressed Sensing. IEEE Transactions on Information Theory, 52, 1289-1306. [Google Scholar] [CrossRef] 
[11] 聂操男. 压缩感知中的概率约束优化模型的光滑近似[D]: [硕士学位论文]. 大连: 辽宁师范大学, 2019.
[12] 陈苗. 一类含有l1范数优化问题的梯度类算法[D]: [硕士学位论文]. 青岛: 青岛大学, 2020.
[13] 屈彪, 张文伟, 于丽超. 线性方程组l1范数问题的松弛投影算法及其应用[J]. 运筹学学报, 2017, 21(2): 57-65.
[14] 王榆. 基于修正的χ2-距离散度的不确定概率优化问题[D]: [硕士学位论文]. 大连: 辽宁师范大学, 2016.
[15] Hu, Z. and Hong, L.J. (2013) Kullback-Leibler Divergence Constrained Distributionally Robust Optimization. Optimization Online, 1, 9.
[16] Shapiro, A., Dentcheva D. and Ruszczyński, A. (2009) Lectures on Stochastic Programming. Society for Industrial and Applied Mathematics. [Google Scholar] [CrossRef] 

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