基于加权截断Schatten-p范数与改进二阶全变分的矩阵填充
Matrix Completion Based on Weighted Truncated Schatten-p Norm and Improved Second Order Total Variation
摘要: 近年来，矩阵填充问题在许多实际应用中引起了研究人员的广泛关注，并产生了一些基于低秩矩阵恢复理论的矩阵填充方法。在这些方法中，人们一般只关注于矩阵的低秩先验信息部分。但是，在矩阵填充的实际应用中，局部光滑先验信息却没有得到更好的利用，使得实际应用中的矩阵填充效果往往不佳。针对上述问题，本文提出了基于加权截断Schatten-p范数与改进二阶全变分的矩阵填充模型。该模型利用加权截断Schatten-p范数对矩阵进行低秩约束，同时利用改进的二阶全变分范数对矩阵的光滑先验信息进行建模，以此来提高矩阵填充效果。通过与多种已有的常用矩阵填充方法在文本掩膜图像重建中的实验结果对比，所提出的方法具有更好的矩阵填充效果。
Abstract: In recent years, matrix completion problem has attracted researchers’ interest in many practical applications. A lot of matrix completion methods based on low rank matrix recovery theory have been developed. In these methods, only the low rank prior information of matrices is considered. However, in the practical application of matrix completion, the local smooth priori information has not been better utilized, which leads to poor matrix completion effect. To solve the above problems, this paper proposes a matrix completion model based on weighted truncated Schatten-p norm and improved second-order total variation. This model uses weighted truncated Schatten-p norm to constrain the matrix with low rank prior. The smooth prior of the matrix is modeled by the improved second-order total variation norm. Compared with the experimental results of many existing matrix completion methods in text mask image reconstruction, the proposed method has better completion effect.
文章引用：陈刚. 基于加权截断Schatten-p范数与改进二阶全变分的矩阵填充[J]. 人工智能与机器人研究, 2019, 8(1): 24-35. https://doi.org/10.12677/AIRR.2019.81004

参考文献

[1] Komodakis. N. (2006) Image Completion Using Global Optimization. Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, New York, 17-22 June 2006, 442-452. [Google Scholar] [CrossRef] 
[2] Keshavan, H. and Montanari, S. (2010) A Matrix Completion from a Few Entries. EEE Transactions on Information Theory, 56, 2980-2998. [Google Scholar] [CrossRef] 
[3] Li, W., Zhao, L., Lin, Z., et al. (2015) Non-Local Image Inpainting Using Low-Rank Matrix Completion. Computer Graphics Forum, 34, 111-122. [Google Scholar] [CrossRef] 
[4] Cand`es, E.J. and Recht, B. (2009) Exact Matrix Completion via Convex Optimization. Foundations of Computational Mathematics, 9, 717-772. [Google Scholar] [CrossRef] 
[5] Yu, T., Yan, J., Liu, W. and Li, B. (2018) Incremental Multi-Graph Matching via Diversity and Randomness Based Graph Clustering. Proceedings of the European Conference on Computer Vision (ECCV), Munich, 8-14 September 2018, 139-154. [Google Scholar] [CrossRef] 
[6] Yu, T. Yan, J., Zhao, J., et al. (2018) Joint Cuts and Matching of Partitions in One Graph. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Salt Lake City, 18-23 June 2018, 705-713. [Google Scholar] [CrossRef] 
[7] Koren, Y., Bell, R. and Volinsky, C. (2009) Matrix Factorization Techniques for Recommender Systems. The Computer Journal, 42, 30-37.
[8] Cabral, R., Del, T., Costeira, P., et al. (2015) Matrix Completion for Weakly-Supervised Multi-Label Image Classification. IEEE Transactions on Pattern Analysis & Machine Intelligence, 37, 121-135. [Google Scholar] [CrossRef] 
[9] Yong, L., Liu, T., Tao, D., et al. (2015) Multiview Matrix Completion for Multilabel Image Classification. IEEE Transactions on Image Processing, 24, 2355-2368. [Google Scholar] [CrossRef] 
[10] Recht, B., Fazel, M. and Parrilo, P.A. (2010) Guaranteed Minimumrank Solu-tions of Linear Matrix Equations via Nuclear Norm Minimization. SIAM Review, 52, 471-501. [Google Scholar] [CrossRef] 
[11] Nie, F., Huang, H. and Ding, C. (2012) Low Rank Matrix Recovery via Efficient Schatten-p Norm Minimization. Proceedings of the 26th AAAI Conference on Artificial Intelligence, Toronto, Ontario, Canada: AAAI Press, 655-661.
[12] Hu, Y., Zhang, D., Ye, J., et al. (2013) Fast and Accurate Matrix Completion via Truncated Nuclear Norm Regularization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35, 2117-2130. [Google Scholar] [CrossRef] 
[13] Oh, T., Tai, Y., Bazin, J., Kim, H. and Kweon, I. (2016) Partial Summinimization of Singular Values in Robust PCA: Algorithm and Applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 38, 744-758. [Google Scholar] [CrossRef] 
[14] Gu, S., Xie, Q., Meng, D., et al. (2017) Weighted Nuclear Norm Minimization and Its Applications to Low Level Vision. International Journal of Computer Vision, 121, 183-208. [Google Scholar] [CrossRef] 
[15] Xie, Y., Gu, S., Liu, Y., et al. (2016) Weighted Schatten p-Norm Minimization for Image Denoising and Background Subtraction. IEEE Transactions on Image Processing, 25, 4842-4857. [Google Scholar] [CrossRef] 
[16] Han, X., Wu, J., Wang, L., Chen, Y. and Shu, H. (2014) Linear Total Variation Approximate Regularized Nuclear Norm Optimization for Matrix Completion. Abstract and Applied Analysis, 2014, Article ID: 765782. [Google Scholar] [CrossRef] 
[17] Wang, W. and Wang, J. (2018) Enhancing Matrix Completion Using a Modified Second-Order Total Variation. Discrete Dynamics in Nature and Society, 2018, Article ID: 2598160. [Google Scholar] [CrossRef] 
[18] Rodriguez, P. (2013) Total Variation Regularization Algorithms Forimages Corrupted with Different Noise Models: A Review. Journal of Electrical and Computer Engineering, 2013, Article ID: 217021.
[19] Papafitsoros, K., Schoenlieb, C.B. and Sengul, B. (2013) Combined First and Second Order Total Variation in Painting Using Split Bregman. Image Processing on Line, 3, 112-136. [Google Scholar] [CrossRef] 
[20] Boyd, S., Parikh, N., Chu, E., Peleato, B. and Eckstein, J. (2011) Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. Foundations and Trends in Machine Learning, 3, 1-122. [Google Scholar] [CrossRef] 
[21] Lin, Z., Liu, R. and Su, Z. (2011) Linearized Alternating Direction Method with Adaptive Penalty for Low-Rank Representation. Proceedings of the 25th Annual Conference on Neural Information Processing Systems, Granada, 12-15 December 2011, 1-7.
[22] Benner, P., Li, R.C. and Truhar, N. (2009) On the ADI Method for Sylvester Equations. Journal of Computational and Applied Mathematics, 233, 1035-1045. [Google Scholar] [CrossRef] 
[23] Lai, M.J., Xu, Y.Y. and Yin, W.T. (2013) Improved Iteratively Reweighted Least Squares for Unconstrained Smoothed l𝑞 Minimization. SIAM Journal on Numerical Analysis, 51, 927-957. [Google Scholar] [CrossRef] 
[24] Wang, Z., Bovik, A.C., Sheikh, H.R. and Simoncelli, E.P. (2004) Image Quality As-sessment: Fromerror Visibility to Structural Similarity. IEEE Transactions on Image Processing, 13, 600-612. [Google Scholar] [CrossRef] 
[25] Yokota, T., Zhao, Q. and Cichocki, A. (2016) Smooth PARAFAC Decomposi-tion for Tensor Completion. IEEE Transactions on Signal Processing, 64, 5423-5436. [Google Scholar] [CrossRef] 

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