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Issue No.01 - Jan. (2013 vol.35)
pp: 208-220
Ji Liu , Univ. of Wisconsin-Madison, Madison, WI, USA
P. Musialski , VRVis Res. Center, Vienna, Austria
P. Wonka , Arizona State Univ., Tempe, AZ, USA
Jieping Ye , Arizona State Univ., Tempe, AZ, USA
In this paper, we propose an algorithm to estimate missing values in tensors of visual data. The values can be missing due to problems in the acquisition process or because the user manually identified unwanted outliers. Our algorithm works even with a small amount of samples and it can propagate structure to fill larger missing regions. Our methodology is built on recent studies about matrix completion using the matrix trace norm. The contribution of our paper is to extend the matrix case to the tensor case by proposing the first definition of the trace norm for tensors and then by building a working algorithm. First, we propose a definition for the tensor trace norm that generalizes the established definition of the matrix trace norm. Second, similarly to matrix completion, the tensor completion is formulated as a convex optimization problem. Unfortunately, the straightforward problem extension is significantly harder to solve than the matrix case because of the dependency among multiple constraints. To tackle this problem, we developed three algorithms: simple low rank tensor completion (SiLRTC), fast low rank tensor completion (FaLRTC), and high accuracy low rank tensor completion (HaLRTC). The SiLRTC algorithm is simple to implement and employs a relaxation technique to separate the dependant relationships and uses the block coordinate descent (BCD) method to achieve a globally optimal solution; the FaLRTC algorithm utilizes a smoothing scheme to transform the original nonsmooth problem into a smooth one and can be used to solve a general tensor trace norm minimization problem; the HaLRTC algorithm applies the alternating direction method of multipliers (ADMMs) to our problem. Our experiments show potential applications of our algorithms and the quantitative evaluation indicates that our methods are more accurate and robust than heuristic approaches. The efficiency comparison indicates that FaLTRC and HaLRTC are more efficient than SiLRTC and between FaLRTC and HaLRTC the former is more efficient to obtain a low accuracy solution and the latter is preferred if a high-accuracy solution is desired.
Tensile stress, Optimization, Minimization, Algorithm design and analysis, Convex functions, Convergence, Smoothing methods,sparse learning, Tensor completion, trace norm
Ji Liu, P. Musialski, P. Wonka, Jieping Ye, "Tensor Completion for Estimating Missing Values in Visual Data", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.35, no. 1, pp. 208-220, Jan. 2013, doi:10.1109/TPAMI.2012.39
[1] Y. Amit, M. Fink, N. Srebro, and S. Ullman, "Uncovering Shared Structures in Multiclass Classification," Proc. 24th Int'l Conf. Machine Learning, pp. 17-24, 2007.
[2] A. Argyriou, T. Evgeniou, and M. Pontil, "Multi-Task Feature Learning," Proc. Advances in Neural Information Processing Systems, pp. 243-272, 2007.
[3] F.R. Bach, "Consistency of Trace Norm Minimization," J. Machine Learning Research, vol. 9, pp. 1019-1048, 2008.
[4] M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester, "Image Inpainting," Proc. ACM Siggraph, pp. 414-424, 2000.
[5] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, "Distributed Optimization and Statistical Learning via the Alternating Direction Method and Multipliers," Foundations and Trends in Machine Learning, vol. 3, pp. 1-122, 2011.
[6] J. Cai, "Fast Singular Value Thresholding without Singular Value Decomposition," UCLA CAM Report, 2010.
[7] J.-F. Cai, E.J. Candès, and Z. Shen, "A Singular Value Thresholding Algorithm for Matrix Completion," SIAM J. Optimization, vol. 20, no. 4, pp. 1956-1982, 2010.
[8] E.J. Candès, X. Li, Y. Ma, and J. Wright, "Robust Principal Component Analysis?" J. ACM, vol. 58, no. 1, pp. 1-37, 2009.
[9] E.J. Candès and B. Recht, "Exact Matrix Completion via Convex Optimization," Foundations of Computational Math., vol. 9, no. 6, pp. 717-772, 2009.
[10] E.J. Candès and T. Tao, "The Power of Convex Relaxation: Near-Optimal Matrix Completion," IEEE Trans. Information Theory, vol. 56, no. 5, pp. 2053-2080, May 2010.
[11] L. Elden, Matrix Methods in Data Mining and Pattern Recognition. SIAM, 2007.
[12] M. Fazel, "Matrix Rank Minimization with Applications," PhD thesis, Stanford Univ., 2002.
[13] M. Fazel, H. Hindi, and S. Boyd, "A Rank Minimization Heuristic with Application to Minimum Order System Approximation," Proc. Am. Control Conf., pp. 4734-4739, 2001.
[14] S. Gandy, B. Recht, and I. Yamada, "Tensor Completion and Low-N-Rank Tensor Recovery via Convex Optimization," Inverse Problem, vol. 27, p. 025010, 2011.
[15] D. Gross, Y.-K. Liu, S.T. Flammia, S. Becker, and J. Eisert, "Quantum State Tomography via Compressed Sensing," Physical Rev. Letters, vol. 105, p. 150401, http//, 2010.
[16] L. Guo, Y. Li, J. Yang, and L. Lu, "Hole-Filling by Rank Sparsity Tensor Decomposition for Medical Imaging," Proc. Int'l Conf. E-Product E-Service and E-Entertainment, pp. 1-4, 2010.
[17] R.A. Harshman, "Foundations of the Parafac Procedure: Models and Conditions for an "Explanatory" Multi-Modal Factor Analysis," UCLA Working Papers in Phonetics, vol. 16, pp. 1-84, 1970.
[18] C.J. Hillar and L. heng Lim, "Most Tensor Problems Are NP Hard," Computing Research Repository,, 2009.
[19] S. Ji, L. Sun, R. Jin, and J. Ye, "Multi-Label Multiple Kernel Learning," Proc. Advances in Neural Information Processing Systems, pp. 777-784, 2008.
[20] S. Ji and J. Ye, "An Accelerated Gradient Method for Trace Norm Minimization," Proc. 26th Ann. Int'l Conf. Machine Learning, pp. 457-464, 2009.
[21] R. Keshavan, A. Montanari, and S. Oh, "Matrix Completion from a Few Entries," IEEE Trans. Information Theory, vol. 56, no. 6, pp. 2980-2998,, June 2010.
[22] N. Komodakis and G. Tziritas, "Image Completion Using Global Optimization," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 417-424, 2006.
[23] T. Korah and C. Rasmussen, "Spatiotemporal Inpainting for Recovering Texture Maps of Occluded Building Facades," IEEE Trans. Image Processing, vol. 16, no. 9, pp. 2262-2271, Sept. 2007.
[24] M. Kurucz, A.A. Benczur, and K. Csalogany, "Methods for Large Scale SVD with Missing Values," Proc. 13th ACM SIGKDD Conf. KDD Cup and Workshop, pp. 31-38, 2007.
[25] N. Li and B. Li, "Tensor Completion for On-Board Compression of Hyperspectral Images," Proc. IEEE 17th Int'l Conf. Image Processing, pp. 517-520, 2010.
[26] Y. Li, J. Yan, Y. Zhou, and J. Yang, "Optimum Subspace Learning and Error Correction for Tensors," Proc. 11th European Conf. Computer Vision, 2010.
[27] Y. Li, Y. Zhou, J. Yan, J. Yang, and X. He, "Tensor Error Correction for Corrupted Values in Visual Data," Proc. IEEE 17th Int'l Conf. Image Processing, pp. 2321-2324, 2010.
[28] Z. Lin, M. Chen, and Y. Ma, "The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices," Technical Report UILU-ENG-09-2215, UIUC (arXiv: 1009.5055), 2009.
[29] J. Liu, P. Musialski, P. Wonka, and J. Ye, "Tensor Completion for Estimating Missing Values in Visual Data," Proc. 12th IEEE Int'l Conf. Computer Vision, pp. 2114-2121, 2009.
[30] S. Ma, D. Goldfarb, and L. Chen, "Fixed Point and Bregman Iterative Methods for Matrix Rank Minimization," Math. Programming, vol. 128, no. 1, pp. 321-353, 2009.
[31] A. Nemirovski, "Efficient Methods in Convex Programming," 1995.
[32] Y. Nesterov, "A Method of Solving a Convex Programing Problem with Convergence Rate $o(1/k^2)$ ," Soviet Math. Doklady, vol. 27, no. 2, pp. 372-376, 1983.
[33] Y. Nesterov, "Introductory Lectures on Convex Programming," Lecture Notes, pp. 119-120, 1998.
[34] Y. Nesterov, "Smooth Minimization of Non-Smooth Functions," Math. Programming, vol. 103, no. 1, pp. 127-152, 2005.
[35] T.K. Pong, P. Tseng, S. Ji, and J. Ye, "Trace Norm Regularization: Reformulations, Algorithms, and Multi-Task Learning," SIAM J. Optimization, vol. 20, no. 6, pp. 3465-3489, 2010.
[36] B. Recht, "A Simpler Approach to Matrix Completion," J. Machine Learning Research, vol. 11, pp. 2287-2322, 2010.
[37] B. Recht, M. Fazel, and P.A. Parrilo, "Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization," Proc. SIAM Rev., vol. 52, no. 3, pp. 471-501, 2010.
[38] M. Signoretto, L.D. Lathauwer, and J.A.K. Suykens, "Nuclear Norms for Tensors and Their Use for Convex Multilinear Estimation," Linear Algebra and Its Applications, submitted, 2010.
[39] N. Srebro, J.D.M. Rennie, and T.S. Jaakkola, "Maximum-Margin Matrix Factorization," Proc. Advances in Neural Information Processing Systems, pp. 1329-1336, 2005.
[40] J.F. Sturm, "Using Sedumi 1.02, a Matlab Toolbox for Optimization over Symmetric Cones," Optimization Methods and Software, vol. 11, pp. 623-625, 1998.
[41] K.C. Toh, M.J. Todd, and R.H. Tutuncu, "SDPT3: A Matlab Software Package for Semidefinite Programming," Optimization Methods and Software, vol. 11, pp. 545-581, 1999.
[42] M. Tomasi and T. Kanade, "Shape and Motion from Image Stream under Orthography: A Factorization Method," Int'l J. Computer Vision, vol. 9, pp. 137-154, 1992.
[43] R. Tomioka, K. Hayashi, and H. Kashima, "Estimation of Low-Rank Tensors via Convex Optimization,", 2011.
[44] O. Troyanskaya, M. Cantor, G. Sherlock, P. Brown, T. Hastie, R. Tibshirani, D. Botstein, and R.B. Altman, "Missing Value Estimation Methods for DNA Microarrays," Bioinformatics, vol. 17, pp. 520-525, 2001.
[45] P. Tseng, "Convergence of Block Coordinate Descent Method for Nondifferentiable Minimization," J. Optimization Theory Application, vol. 109, pp. 475-494, 2001.
[46] L.R. Tucker, "Some Mathematical Notes on Three-Mode Factor Analysis," Psychometrika, vol. 31, pp. 279-311, 1966.
[47] J. Yan, J. Liu, Y. Li, Z. Niu, and Y. Liu, "Visual Saliency Detection via Rank-Sparsity Decomposition," Proc. 17th IEEE Int'l Conf. Image Processing, pp. 1089-1092, 2010.
[48] Z. Zhou, X. Li, J. Wright, E.J. Candès, and Y. Ma, "Stable Principal Component Pursuit," Computing Research Repository,, 2010.
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