Truncated nuclear norm minimization for tensor completion

In this paper, a tensor n-mode matrix unfolding truncated nuclear norm is proposed, which is extended from the matrix truncated nuclear norm, to tensor completion problem. The alternating direction method of multipliers is utilized to solve this optimization problem. Meanwhile, the original two-step solution of the matrix truncated nuclear norm is reduced to one step. Employing the intermediate results returned by singular value shrinkage operator, rank information of each tensor unfolding matrix is not required and thus the computational complexity of the devised approach is not demanding. Computer simulation results demonstrate the effectiveness of the proposed method.

[1]  Xiangjian He,et al.  Tensor error correction for corrupted values in visual data , 2010, 2010 IEEE International Conference on Image Processing.

[2]  Emmanuel J. Candès,et al.  The Power of Convex Relaxation: Near-Optimal Matrix Completion , 2009, IEEE Transactions on Information Theory.

[3]  Emmanuel J. Candès,et al.  A Singular Value Thresholding Algorithm for Matrix Completion , 2008, SIAM J. Optim..

[4]  Gonzalo Mateos,et al.  Rank Regularization and Bayesian Inference for Tensor Completion and Extrapolation , 2013, IEEE Transactions on Signal Processing.

[5]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[6]  Jieping Ye,et al.  Tensor Completion for Estimating Missing Values in Visual Data , 2009, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[7]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[8]  Olgica Milenkovic,et al.  A Geometric Approach to Low-Rank Matrix Completion , 2010, IEEE Transactions on Information Theory.

[9]  Dave Zachariah,et al.  Alternating Least-Squares for Low-Rank Matrix Reconstruction , 2012, IEEE Signal Processing Letters.

[10]  Xianjun Shi,et al.  A Fixed Point Iterative Method for Low $n$-Rank Tensor Pursuit , 2013, IEEE Transactions on Signal Processing.

[11]  Christopher J. Hillar,et al.  Most Tensor Problems Are NP-Hard , 2009, JACM.

[12]  Johan A. K. Suykens,et al.  Tensor Versus Matrix Completion: A Comparison With Application to Spectral Data , 2011, IEEE Signal Processing Letters.

[13]  Ieee Staff.  2014 IEEE 8th Sensor Array and Multichannel Signal Processing Workshop (SAM) , 2014 .

[14]  Yuanyuan Liu,et al.  An Efficient Matrix Factorization Method for Tensor Completion , 2013, IEEE Signal Processing Letters.

[15]  Bart Vandereycken,et al.  Low-rank tensor completion by Riemannian optimization , 2014 .

[16]  Xuelong Li,et al.  Fast and Accurate Matrix Completion via Truncated Nuclear Norm Regularization , 2013, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[17]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2008, Found. Comput. Math..

[18]  B. Recht,et al.  Tensor completion and low-n-rank tensor recovery via convex optimization , 2011 .