删除或更新信息,请邮件至freekaoyan#163.com(#换成@)

Toeplitz矩阵填充的四种流形逼近算法比较

本站小编 Free考研考试/2021-12-27

韩如意1, 王川龙2
1. 太原理工大学数学学院, 太原 030024;
2. 太原师范学院工程科学计算山西省高等学校重点实验室, 太原 030619
收稿日期:2017-09-07出版日期:2018-09-15发布日期:2018-08-08
通讯作者:王川龙,E-mail:clwang1964@163.com.

基金资助:国家自然科学基金(11371275)和山西省自然科学基金(201601D011004)资助.


THE COMPARISON OF FOUR MANIFOLD APPROXIMATION ALGORITHMS FOR TOEPLITZ MATRIX COMPLETION

Han Ruyi1, Wang Chuanlong2
1. Institution of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China;
2. Higher Education Key Laboratory of Engineering Science Computing in Shanxi Province, Taiyuan Normal University, Taiyuan 030619, China
Received:2017-09-07Online:2018-09-15Published:2018-08-08







摘要



编辑推荐
-->


本文提出Toeplitz矩阵填充的四种流形逼近算法。在左奇异向量空间中对已知部分运用最小二乘法逼近,形成新的可行矩阵;并将对角线上的元素分别用均值,l1范数,l范数和中间数四种方法逼近使得迭代后的矩阵仍保持Toeplitz结构,节约了奇异向量空间的分解时间。最终找到合理的低秩矩阵来逼近未知的高秩矩阵,进而精确地完成Toeplitz矩阵的填充。理论上,分析了在一定条件下算法的收敛性。实验上,通过取不同的采样密度进行数值实验展示了四种算法的优劣。实验结果说明均值算法和l范数算法大多用的时间较少,但是当采样密度和矩阵规模较大时,中间数算法的精度较高。
MR(2010)主题分类:
15A83
90C06

分享此文:


()

[1] Bennett J,Lanning S.The netflix prize[J].In Proceedings of KDD Cup and Workshop,2007.

[2] Bertalmio M,Sapiro G,Caselles V,Ballester C.Image inpainting[J].Comput.Grapher,(SIGGRAPH 2000),2000,34:417-424.

[3] Argyriou A,Evgeniou T,Pontil M.Multi-task feature learing[J].Adv.Neural Information Processing Systems,2007,19:41-48.

[4] Tomasi C,Kanade T.Shape and motion from image streams under orthography:a factorization method[J].Int.J.Comput Vision,1992,9:137-154.

[5] Mesbahi M,Papavassilopoulos G P.On the rank minimization problem over a positive semidefinite linear matrix inequality[J].IEEE Transactions on Automatic Control,1997,42:239-243.

[6] Moon Y S,Park P G,Kwon W H.Delay-dependent robust stabilization of uncertain state-delayed systems[J].Internation Journal of Control,2001,74(14):1447-1455.

[7] Candès E J,Plan Y.Matrix completion with noise[J].Proceedings of the IEEE,2010,98(6):925-936.

[8] Chen P,Suter D.Recovering the missing components in a large noisy low-rank matrix:application to sfm source[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,2004,26:1051-1063.

[9] Abernethy F,Evgeniou T,Vert J P.A new approach to collaborative filtering:operator estimation with spectral regularization[J].Journal of Machine Learning Research,2009,10:803-826.

[10] Nicolas Boumal,Absil P A.RTRMC:A Riemannian trust-region method for low-rank matrix completion[A].Annual conference on Neural Information Processing Systems 25th,2011,24:406-414.

[11] Keshavan R H,Montanari A,Sewoong O.Matrix Completion From a Few Entries[J].IEEE Trans.Inform.Theory,2010,56(6):2980-2998.

[12] Lee K,Bresler Y.ADMiRA:Atomic Decomposition for Minimum Rank Approximation[J].IEEE Trans.Inform.Theory,2010,56(9):4402-4416.

[13] Candès E J,Recht B.Exact matrix completion via convex optimization[J].Found.Comput.Math.,2009,9(6):717-772.

[14] Toh K C,Yun S.An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems[J].Pacific J.Optimi.,2010,6:615-640.

[15] Cai J F,Candès E J,Shen Z.A singuar value thresholding algorithm for matrix completion[J].SIAM J.Optim.,2010,20:1956-1982.

[16] Lin Z,Chen M,Wu L,Ma Y.The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices[M].UIUC Technicial Report UIUL-ENG-09-2214,2010.

[17] Lin Z,Ganesh A,Wright J,Wu L,Chen M,Ma Y.Fast convex optimization algorithms for exact recovery of a corrupted low-rank matrix[A].IEEE International Workshop on Computational Advances in Multi-sensor Adaptive Processing,Aruba,Dutch Antilles,2009,56(3):707-722.

[18] Dai W,Milenkovic O.SET:an algorithm for consistent matrix completion[A].IEEE International Conference on Acoustics,Speech,and Signal Processing,2010,3646-3649.

[19] Ma S,Goldfarb D,Chen L.Fixed point and Bregman iterative methods for matrix rank minimization[J].Math.Program.,2011,128(1-2):321-353.

[20] Chen C,He B,Yuan X.Matrix completion via an alternating direction method[J].Ima.J.Numer.Anal.,2012,32(1):227-245.

[21] Yuan M,Joseph R,Zou H.Structured variable selection and estimation[J].Annals of Applied Statistics,2009,3(4):1738-1757.

[22] Ma Y,Zhi L.The minimum-rank gram matrix completion via modified fixed point continuation method[A].International Symposium on Symbolic and Algebraic Computation 36th,2011,241-248.

[23] Jain P,Netrapalli P,Sanghavi S.Low-rank matrix completion using alternating minimization[J].Statistics,2012,665-674.

[24] 刘丽霞,王川龙.基于均值的Toeplitz矩阵填充的子空间算法[J].计算数学,2017,2(39):179-188.

[1]吴敏华, 李郴良. 求解带Toeplitz矩阵的线性互补问题的一类预处理模系矩阵分裂迭代法[J]. 计算数学, 2020, 42(2): 223-236.
[2]刘丽霞, 王川龙. 基于均值的Toeplitz矩阵填充的子空间算法[J]. 计算数学, 2017, 39(2): 179-188.
[3]刘仲云, 刘成志, 张育林. 对称正定Toeplitz方程组的多级迭代求解[J]. 计算数学, 2012, 34(4): 397-404.
[4]袁永新,戴华. 线性流形上的广义中心对称矩阵反问题[J]. 计算数学, 2005, 27(4): 383-394.
[5]周富照,胡锡炎,张磊. 线性流形上对称正交对称矩阵逆特征值问题[J]. 计算数学, 2003, 25(3): 281-292.
[6]张忠志,胡锡炎,张磊. 线性流形上Hermite-广义反Hamilton矩阵反问题的最小二乘解[J]. 计算数学, 2003, 25(2): 209-218.
[7]黄艾香,刘之行,张引娣. 基于近似惯性流形的后验Galerkin的方法[J]. 计算数学, 2001, 23(3): 289-298.
[8]张磊,谢冬秀,胡锡炎. 线性流形上双对称阵逆特征值问题[J]. 计算数学, 2000, 22(2): 129-138.
[9]李开泰,侯延仁. N-S方程一般近似惯性流形构造和逼近[J]. 计算数学, 1999, 21(3): 269-282.
[10]廖安平. 线性流形上矩阵方程AX=B的一类反问题及数值解法[J]. 计算数学, 1998, 20(4): 371-376.

--> -->
阅读次数
全文







摘要





Cited

Shared






PDF全文下载地址:

http://www.computmath.com/jssx/CN/article/downloadArticleFile.do?attachType=PDF&id=242
相关话题/计算 数学 空间 实验 矩阵