A Finite Algorithm to Compute Rank-1 Tensor Approximations

We propose a noniterative algorithm, called SeROAP,1 to estimate a rank-1 approximation of a tensor in the real or complex field. Our algorithm is based on a sequence of singular value decompositions followed by a sequence of projections onto Kronecker vectors. For three-way tensors, we show that our algorithm is always at least as good as the state-of-the-art truncation algorithm, ST-HOSVD,2 in terms of approximation error. Thus, it gives a good starting point to iterative rank-1 tensor approximation algorithms. By means of computational experiments, it also turns out that for fourth order tensors, SeROAP yields a better approximation with high probability when compared to the standard THOSVD3 algorithm.

[1]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[2]  H. Kiers.  Towards a standardized notation and terminology in multiway analysis , 2000 .

[3]  Bernard Mourrain,et al.  Border basis relaxation for polynomial optimization , 2016, J. Symb. Comput..

[4]  Souleymen Sahnoun,et al.  Joint Source Estimation and Localization , 2015, IEEE Transactions on Signal Processing.

[5]  Joos Vandewalle,et al.  A Multilinear Singular Value Decomposition , 2000, SIAM J. Matrix Anal. Appl..

[6]  Lieven De Lathauwer,et al.  Optimization-Based Algorithms for Tensor Decompositions: Canonical Polyadic Decomposition, Decomposition in Rank-(Lr, Lr, 1) Terms, and a New Generalization , 2013, SIAM J. Optim..

[7]  Li Wang,et al.  Semidefinite Relaxations for Best Rank-1 Tensor Approximations , 2013, SIAM J. Matrix Anal. Appl..

[8]  Pierre Comon,et al.  Tensors : A brief introduction , 2014, IEEE Signal Processing Magazine.

[9]  P. Paatero.  A weighted non-negative least squares algorithm for three-way ‘PARAFAC’ factor analysis , 1997 .

[10]  Vin de Silva,et al.  Tensor rank and the ill-posedness of the best low-rank approximation problem , 2006, math/0607647.

[11]  André Uschmajew,et al.  Local Convergence of the Alternating Least Squares Algorithm for Canonical Tensor Approximation , 2012, SIAM J. Matrix Anal. Appl..

[12]  Eric Moreau,et al.  A generalization of joint-diagonalization criteria for source separation , 2001, IEEE Trans. Signal Process..

[13]  G. Golub,et al.  Tracking a few extreme singular values and vectors in signal processing , 1990, Proc. IEEE.

[14]  Laurent Albera,et al.  Joint Eigenvalue Decomposition of Non-Defective Matrices Based on the LU Factorization With Application to ICA , 2015, IEEE Transactions on Signal Processing.

[15]  Berkant Savas,et al.  Algorithms in data mining using matrix and tensor methods , 2008 .

[16]  Raf Vandebril,et al.  A New Truncation Strategy for the Higher-Order Singular Value Decomposition , 2012, SIAM J. Sci. Comput..

[17]  J. Landsberg.  Tensors: Geometry and Applications , 2011 .

[18]  J. Berge,et al.  Kruskal's polynomial for 2×2×2 arrays and a generalization to 2×n×n arrays , 1991 .

[19]  Luke Oeding,et al.  Eigenvectors of tensors and algorithms for Waring decomposition , 2011, J. Symb. Comput..

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

[21]  Pierre Comon,et al.  Handbook of Blind Source Separation: Independent Component Analysis and Applications , 2010 .

[22]  YUNING YANG,et al.  Rank-1 Tensor Properties with Applications to a Class of Tensor Optimization Problems , 2016, SIAM J. Optim..

[23]  Pierre Comon,et al.  Symmetric tensor decomposition , 2009, 2009 17th European Signal Processing Conference.

[24]  André Lima Férrer de Almeida,et al.  PARAFAC-based unified tensor modeling for wireless communication systems with application to blind multiuser equalization , 2007, Signal Process..

[25]  X. Luciani,et al.  Canonical Polyadic Decomposition based on joint eigenvalue decomposition , 2014 .

[26]  F Wendling,et al.  EEG extended source localization: Tensor-based vs. conventional methods , 2014, NeuroImage.

[27]  Haibin Ling,et al.  Multi-target Tracking by Rank-1 Tensor Approximation , 2013, 2013 IEEE Conference on Computer Vision and Pattern Recognition.

[28]  P. Comon,et al.  Tensor decompositions, alternating least squares and other tales , 2009 .

[29]  Eric Moreau,et al.  Optimal Joint Diagonalization of Complex Symmetric Third-Order Tensors. Application to Separation of Non Circular Signals , 2007, ICA.

[30]  Pierre Comon,et al.  Enhanced Line Search: A Novel Method to Accelerate PARAFAC , 2008, SIAM J. Matrix Anal. Appl..

[31]  Gene H. Golub,et al.  Rank-One Approximation to High Order Tensors , 2001, SIAM J. Matrix Anal. Appl..

[32]  Lieven De Lathauwer,et al.  A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization , 2006, SIAM J. Matrix Anal. Appl..

[33]  W. Hackbusch.  Tensor Spaces and Numerical Tensor Calculus , 2012, Springer Series in Computational Mathematics.

[34]  Rasmus Bro,et al.  A comparison of algorithms for fitting the PARAFAC model , 2006, Comput. Stat. Data Anal..

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

[36]  Pierre Comon,et al.  Rank-1 Tensor Approximation Methods and Application to Deflation , 2015, ArXiv.

[37]  David E. Booth,et al.  Multi-Way Analysis: Applications in the Chemical Sciences , 2005, Technometrics.

[38]  Chris Peterson,et al.  Induction for secant varieties of Segre varieties , 2006, math/0607191.