publication . Other literature type . Article . Preprint . 2011

A Nonlinear GMRES Optimization Algorithm for Canonical Tensor Decomposition

De Sterck, Hans;
  • Published: 26 May 2011
  • Publisher: Society for Industrial & Applied Mathematics (SIAM)
Abstract
A new algorithm is presented for computing a canonical rank-R tensor approximation that has minimal distance to a given tensor in the Frobenius norm, where the canonical rank-R tensor consists of the sum of R rank-one components. Each iteration of the method consists of three steps. In the first step, a tentative new iterate is generated by a stand-alone one-step process, for which we use alternating least squares (ALS). In the second step, an accelerated iterate is generated by a nonlinear generalized minimal residual (GMRES) approach, recombining previous iterates in an optimal way, and essentially using the stand-alone one-step process as a preconditioner. In...
Subjects
arXiv: Computer Science::Numerical Analysis
free text keywords: Applied Mathematics, Computational Mathematics, Preconditioner, Nonlinear system, Matrix norm, Generalized minimal residual method, Iterated function, Line search, Tensor, Mathematics, Mathematical analysis, Nonlinear programming, Mathematics - Numerical Analysis, Computer Science - Numerical Analysis
18 references, page 1 of 2

[1] E. Acar, D.M. Dunlavy, and T.G. Kolda, A Scalable Optimization Approach for Fitting Canonical Tensor Decompositions, Journal of Chemometrics, 25 (2011), pp. 67-86. [OpenAIRE]

[2] B.W. Bader and T.G. Kolda, MATLAB Tensor Toolbox Version 2.4, http://csmr.ca.sandia.gov/∼tgkolda/TensorToolbox/, March 2010.

[3] J.D. Carroll and J.J. Chang, Analysis of individual differences in multidimensional scaling via an N-way generalization of Eckart-Young decomposition, Psychometrika, 35 (1970), pp. 283-319.

[4] L. De Lathauwer, A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization, SIAM Journal on Matrix Analysis and Applications, 28 (2006), pp. 642-666. [OpenAIRE]

[5] L. De Lathauwer, B. De Moor, and J. Vandewalle, Computation of the canonical decomposition by means of a simultaneous generalized Schur decomposition, SIAM Journal on Matrix Analysis and Applications, 26 (2004), pp. 295-327. [OpenAIRE]

[6] D.M. Dunlavy, T.G. Kolda, and E. Acar, Poblano v1.0: A Matlab Toolbox for GradientBased Optimization, Technical Report SAND2010-1422, Sandia National Laboratories, Albuquerque, NM and Livermore, CA, March 2010.

[7] R.A. Harshman, Foundations of the PARAFAC procedure: Models and conditions for an explanatory multi-modal factor analysis, UCLA working papers in phonetics, 16 (1970), pp. 1-84.

[8] M. Ishteva, L. De Lathauwer, P. Absil, and S. Van Huffel, Differential-geometric Newton method for the best rank-(R1 ,R2,R3) approximation of tensors, Numerical Algorithms, 51 (2009), pp. 179-194.

[9] T.G. Kolda and B.W. Bader, Tensor Decompositions and Applications, SIAM Review, 51 (2009), pp. 455-500.

[10] J.J. Mor´e and D.J. Thuente, Line search algorithms with guaranteed sufficient decrease, ACM Transactions on Mathematical Software, 20 (1994), pp. 286-307.

[11] J. Nocedal and S.J. Wright, Numerical optimization, Springer, Berlin, 2006.

[12] C.W. Oosterlee, On multigrid for linear complementarity problems with application to American-style options, Electronic Transactions on Numerical Analysis, 15 (2003), pp. 165-185. [OpenAIRE]

[13] C.W. Oosterlee and T. Washio, Krylov Subspace Acceleration of Nonlinear Multigrid with Application to Recirculating Flows, SIAM J. Sci. Comput., 21 (2000), pp. 1670-1690. [OpenAIRE]

[14] M. Rajih, P. Comon, and RA Harshman, Enhanced line search: A novel method to accelerate PARAFAC, SIAM Journal on Matrix Analysis and Applications, 30 (2008), pp. 1128-1147.

[15] Y. Saad and M.H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Comp., 7 (1986), pp. 856-869.

18 references, page 1 of 2
Powered by OpenAIRE Research Graph
Any information missing or wrong?Report an Issue