固定时间梯度流在l1-l2范数中的稀疏重构

胡登洲; 何兴

doi:10.21656/1000-0887.400202

固定时间梯度流在l₁-l₂范数中的稀疏重构

doi: 10.21656/1000-0887.400202

胡登洲,
何兴

西南大学电子信息工程学院, 重庆 400700

基金项目: 国家自然科学基金(61773320)

详细信息

作者简介:
胡登洲(1992—)，男，硕士生(E-mail: 1076060236@qq.com)；何兴(1986—)，男，教授，博士生导师(通讯作者. E-mail: hexingdoc@swu.edu.cn).

中图分类号: O357.41
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- 被引次数: 0
出版历程
- 收稿日期: 2019-07-02
- 修回日期: 2019-07-09
- 刊出日期: 2019-11-01

Sparse Reconstruction of Fixed-Time Gradient Flow in the l₁-l₂ Norm

College of Electronic and Information Engineering, Southwest University, Chongqing 400700, P.R.China

Funds: The National Natural Science Foundation of China(61773320)

摘要

摘要: 压缩感知（compressed sensing，CS）是一种全新的信号采样技术，对于稀疏信号，它能够以远小于传统的Nyquist采样定理的采样点来重构信号。在压缩感知中，采用动态连续系统，对l₁-l₂范数的稀疏信号重构问题进行了研究。提出了一种基于固定时间梯度流的稀疏信号重构算法，证明了该算法在Lyapunov意义上的稳定性并且收敛于问题的最优解。最后通过与现有的投影神经网络算法的对比，体现了该算法的可行性以及在收敛速度上的优势.
- 压缩感知 /
- l₁-l₂范数 /
- 固定时间梯度流 /
- 稀疏信号重构
Abstract: The compressed sensing (CS) is a new signal sampling technology, which can reconstruct signals at sampling points far smaller than those in the traditional Nyquist sampling theorem for sparse signals. For the compressed sensing, a dynamic continuous system was used to study the sparse signal reconstruction of the l₁l-l₂ norm. A sparse signal reconstruction algorithm based on the fixed time gradient flow was proposed, and was proved to be stable in the sense of Lyapunov and to converge to the optimal solution of the problem. Finally, the feasibility and advantages in the convergence speed of this algorithm were demonstrated through comparison between the proposed algorithm and existing projection neural network algorithms.
- compressed sensing /
- l₁-l₂ norm /
- fixed-time gradient flow /
- sparse signal reconstruction

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参考文献(26)

[1]	CANDS E J. Compressive sampling[J]. Proceedings of International Congress of Mathematicians,2006,17(2): 1433-1452.
[2]	DONOHO D. Compressed sensing[J]. IEEE Transactions on Information Theory,2006,52(4): 1289-1306.
[3]	COMBETTES P L. Signal recovery by proximal forward-backward splitting[J]. Multiscale Modeling & Simulation,2005, 4(4): 1168-1200.
[4]	ZHU J, ROSSET S, HASTIE T. 1-norm support vector machines[C]// Advances in Neural Information Processing Systems . Vancouver, Canada: MIT Press, 2003: 49-56.
[5]	NATARAJAN B K. Sparse approximation solutions to linear systems[J]. SIAM Journal on Computing,1995,24(2): 227-234.
[6]	YIN P, LOU Y, HE Q, et al. Minimization of l₁-l₂ for compressed sensing[J]. SIAM Journal on Scientific Computing,2015,37(1): 536-563.
[7]	FIGUEIREDO M A T, NOWAK R D, WRIGHT S J. Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems[J]. IEEE Journal of Selected Topics in Signal Processing,2007,1(4): 586-597.
[8]	CANDES E, TAO T. Decoding by linear programming[J]. IEEE Transactions on Information Theory,2005, 51(12): 4203-4215.
[9]	YANG A, GANESH A, SASTRY S, et al. Fast l₁-minimization algorithms and an application in robust face recognition[C]//Proceedings of the 17th IEEE International Conference on Image Processing (ICIP) . Hong Kong, China, 2010.
[10]	ANDRES A M, PADOVANI S, TEPPER M, et al. Face recognition on partially occluded images using compressed sensing[J]. Pattern Recognition Letters,2014,36: 235-242.
[11]	MALLAT S. A Wavelet Tour of Signal Processing: the Sparse Way [M]. Burlington: Academic Press, 2009.
[12]	BRUCKSTEIN A M, DONOHO D L, ELAD M. From sparse solutions of systems of equations to sparse modeling of signals and images[J]. SIAM Review,2009,51: 34-81.
[13]	陈鹏清, 黄尉. 基于l1-l2范数的块稀疏信号重构[J]. 应用数学和力学, 2017,38(8): 932-942.(CHEN Pengqing, HUANG Wei. Block-sparse signal recovery based on l₁-l₂ norm minimization[J]. Applied Mathematice and Mechanics,2017,38(8): 932-942.(in Chinese))
[14]	CHEN S S, DONOHO D L, SAUNDERS M A. Atomic decomposition by basis pursuit[J]. SIAM Review,2001, 43(1): 129-159.
[15]	KIM S J, KOH K, LUSTIG M, et al. An interior-point method for large-scale l₁-regularized least squares[J]. IEEE Journal on Selected Topics in Signal Processing,2007,51: 606-617.
[16]	TANK D W, HOPFIELD J J. Simple neural optimization network: an A/D converter, signal decision circuit and a linear programming circuit[J]. IEEE Transactions on Circuits and Systems,1986,33(5): 533-541.
[17]	BALAVOINE A, ROMBERG J, ROZELL C J. Convergence and rate analysis of neural networks for sparse approximation[J]. IEEE Transactions on Neural Networks and Learning Systems,2012,23(9): 1377-1389.
[18]	LIU Q, WANG J. L₁-minimization algorithms for sparse signal reconstruction based on a projection neural network[J]. IEEE Transactions on Neural Networks and Learning Systems,2016,27(3): 698-707.
[19]	LIU Y W, HU J F. A neural network for l₁-l₂ minimization based on scaled gradient projection: application to compressed sensing[J]. Neurocomputing,2016,173(3): 988-993.
[20]	GARG K, PANAGOU D. A new scheme of gradient flow and saddle-point dynamics with fixed-time convergence guarantees[R/OL]. [2019-07-02]. https:/arXiv.org/abs/1808.10474v1.
[21]	FUCHS J J. Multipath time-delay detection and estimation[J]. IEEE Transactions on Signal Processing,1999,47(1): 237-243.
[22]	POLYAKOV A. Nonlinear feedback design for fixed-time stabilization of linear control systems[J]. IEEE Transactions on Automatic Control,2012,57(8): 2106-2110.
[23]	BHAT S P, BERNSTEIN D S. Finite-time stability of continuous autonomous systems[J]. SIAM Journal of Control and Optimization,2000,38(3): 751-766.
[24]	ORLOV Y. Finite time stability and robust control synthesis of uncertain switched systems[J]. SIAM Journal of Control and Optimization,2005,43(4): 1253-1271.
[25]	LOPEZ-RAMIREZ F, EFIMOV D, POLYAKOV A, et al. On necessary and sufficient conditions for fixed-time stability of continuous autonomous systems[C]//17th European Control Conference (ECC) . Cyprus, 2018.
[26]	LI C, YU X, HUANG T, et al. Distributed optimal consensus over resource allocation network and its application to dynamical economic dispatch[J]. IEEE Transactions on Neural Networks and Learning Systems,2018,29(6): 2407-2417.