l_p范数最小化问题下部分稀疏信号的稳定恢复（英文）
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摘要
通过l_p范数最小化模型,研究了在有噪音线性测量值下稳定恢复部分稀疏信号的问题.首先提出了恢复信号的充分条件:部分p-限制等距条件(p-RIP),并推导出此模型的最优解与要恢复的原始信号误差范围.最后在无噪音l_p范数最小化问题模型下,计算出至少多少随机高斯测量值能够以高概率恢复部分稀疏信号.
The problem of recovering a partially sparse signal stably from a given set of noisy linear measurements is studied via l_p norm minimization method. A sufficient partial restricted p-isometry properties(p-RIP) condition is proposed and an error between the solution of the noisy l_p minimization and the original signal needs to recover is obtained. Moreover, in the condition of no noise, a lower bound on the number of random Gaussian measurements is given to recover the partially sparse signal by l_p minimization with high probability.
The problem of recovering a partially sparse signal stably from a given set of noisy linear measurements is studied via l_p norm minimization method. A sufficient partial restricted p-isometry properties(p-RIP) condition is proposed and an error between the solution of the noisy l_p minimization and the original signal needs to recover is obtained. Moreover, in the condition of no noise, a lower bound on the number of random Gaussian measurements is given to recover the partially sparse signal by l_p minimization with high probability.
引文
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14 Chartrand R,Staneva V.Restricted isometry porperties and nonconvex compressive sensing[J].Inverse Problems,2008,24(3):657-682.
15 Shen Y,Li S.Restricted p-isometry property and its application for nonconvex compressive sensing[J].Advances in Computational Mathematics,2010,37(3):441-452.
16 Zonoobi D,Kassim A A.Low rank and sparse matrix reconstruction with partial support knowledge for surveillance video processing[J].IEEE International Conference on Image Processing,2013:335-339.
17 Wan L,Zou C,Zhang H.Full and partial shape similarity through sparse descriptor reconstruction[J].Visual Computer,2016,34:1-13.
18 Bandeira A S,Scheinberg K,Vicente L N.On partially sparse recovery[EB/OL].(2013-04-09)[2017-05-10]http://arxiv.org/abs/1304.2809ar Xiv preprint ar Xiv.
19 Bandeira A S,Scheinberg K,Vicente L N.Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization[J].Mathematical Programming,2013,134(1):223-257.
20 Vaswani N,Lu W.Modified CS:Modifying compressive sensing for problems with partially known support[J].IEEE Transactions on Signal Processing,2010,58(9):4 595-4 607.
21 Bi H,Kong L,Xiu N.Restricted p-isometry properties of partial sparse signal recovery[J].Discrete Dynamics in Nature and Society,2013,12:87-118.
22 Wang Y,Wang J J,Xu Z B.Restricted p-isometry properties of nonconvex blocksparse compressed sensing[J].Signal Processing,2014,104:188-196.
23 Foucart S,Lai M J.Sparsest solutions of underdetermined linear systems via lpminimization for 0
2 Cands E J,Romberg J,Tao T.Stable signal recovery from incomplete and inaccurate measurements[J].Communications on Pure and Applied Mathematics,2006,59(8):1 207-1 223.
3 Cands E J,Romberg J,Tao T.Robust uncertainty principles:exact signal reconstruction from highly incomplete frequency information[J].IEEE Transactions on Information Theory,2006,52(2):489-509.
4 Rossi M.Spatial compressive sensing for MIMO radar[J].IEEE Transactions on Signal Processing,2013,62(2):419-430.
5 Rao M S,Naik K K,Reddy K M.Compressed sensing:Radar signal detection and parameter measurement for Ew applications[J].Radio Science,51(9):1 457-1 469.
6 Li Feng,Leon Axel,Hersh Chandarana,et al.XDGRASP:Golden-angle radial MRI with reconstruction of extra motionstate dimensions using compressed sensing[J].Magnetic Resonance in Medicine,2016,75(2):775-788.
7 Rosenkrantz A B,Geppert C,Grimm R.Dynamic contrast-enhanced MRI of the prostate with high spatiotemporal resolution using compressed sensing,parallel imaging,and continuous golden-angle radial sampling:preliminary experience[J].Journal of Magnetic Resonance Imaging,2015,41(5):1 365-1 373.
8 Leary R,Saghi Z,Midgley P A,et al.Compressed sensing electron tomography[J].Ultramicroscopy,2013,131(8):70-91.
9 Carrillo R E,Mcewen J D,Wiaux Y.PURIFY:a new approach to radiointerferometric imaging[J].Monthly Notices of the Royal Astronomical Society,2014,439(4):3 591-3 604.
10 Donoho D,Elad M.Optimally sparse representation in general(nonorthogonal)dictionaries via l1minimization[J].Proc Natl Acad Sci USA,2003,100(5):2 197-2 202.
11 Natarajan B K.Sparse approxiamte solutions to linear systems[J].Siam Journal on Computing,1995,24(2):227-234.
12 Chartrand R.Nonconvex compressed sensing and error correction[J].IEEE International Conference on Acoustics,2007,3:889-892.
13 Chartrand R.Exact reconstruction of sparse signals via nonconvex minimization[J].IEEE Signal Processing Letters,2007,14(10):707-710.
14 Chartrand R,Staneva V.Restricted isometry porperties and nonconvex compressive sensing[J].Inverse Problems,2008,24(3):657-682.
15 Shen Y,Li S.Restricted p-isometry property and its application for nonconvex compressive sensing[J].Advances in Computational Mathematics,2010,37(3):441-452.
16 Zonoobi D,Kassim A A.Low rank and sparse matrix reconstruction with partial support knowledge for surveillance video processing[J].IEEE International Conference on Image Processing,2013:335-339.
17 Wan L,Zou C,Zhang H.Full and partial shape similarity through sparse descriptor reconstruction[J].Visual Computer,2016,34:1-13.
18 Bandeira A S,Scheinberg K,Vicente L N.On partially sparse recovery[EB/OL].(2013-04-09)[2017-05-10]http://arxiv.org/abs/1304.2809ar Xiv preprint ar Xiv.
19 Bandeira A S,Scheinberg K,Vicente L N.Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization[J].Mathematical Programming,2013,134(1):223-257.
20 Vaswani N,Lu W.Modified CS:Modifying compressive sensing for problems with partially known support[J].IEEE Transactions on Signal Processing,2010,58(9):4 595-4 607.
21 Bi H,Kong L,Xiu N.Restricted p-isometry properties of partial sparse signal recovery[J].Discrete Dynamics in Nature and Society,2013,12:87-118.
22 Wang Y,Wang J J,Xu Z B.Restricted p-isometry properties of nonconvex blocksparse compressed sensing[J].Signal Processing,2014,104:188-196.
23 Foucart S,Lai M J.Sparsest solutions of underdetermined linear systems via lpminimization for 0