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Abstract
The multilinear least-squares (MLLS) problem is an extension of the linear least-squares problem. The difference is that a multilinear operator is used in place of a matrix-vector product. The MLLS is typically a large-scale problem characterized by a large number of local minimizers. It originates, for instance, from the design of filter networks. We present a global search strategy that allows for moving from one local minimizer to a better one. The efficiency of this strategy is illustrated by the results of numerical experiments performed for some problems related to the design of filter networks.
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References
- ANDERSSON, M., WIKLUND, J. and KNUTSSON, H. 1999. Filter networks. In N.M. NAMAZI, editor, Signal and Image Processing (SIP), Proceedings of the IASTED International Conferences, October 18-21, 1999, Nassau, The Bahamas, pages 213–217. IASTED/ACTA Press.
- BJÖRK, Å. 1996. Numerical Methods for Least Squares Problems. SIAM, Philadelphia, USA.
- GRANLUND, G. and KNUTSSON, H. 1995. Signal Processing for Computer Vision. Kluwer, Dordrecht.
- KNUTSSON, H. 1982. Filtering and Reconstruction in Image Processing. PhD thesis, Linköping University, Sweden, Diss. No. 88.
- KNUTSSON, H. and ANDERSSON, M. 2005. Implications of invariance and uncertainty for local structure analysis filter sets. Signal Processing: Image Communications, 20(6):569–581.
- KNUTSSON, H., WESTIN, C-F. and ANDERSSON, M. 2011. Representing local structure using tensors II. In A. HEYDEN and F. KAHL, Editors, Image Analysis, Lecture Notes in Computer Science, 6688: 545–556. Springer, Berlin/Heidelberg.
- LEARDI, R., ARMANINO, C., LANTERI, S. and ALBEROTANZA, L. 2000. Three-mode principal component analysis of monitoring data from Venice lagoon. Journal of Chemometrics, 14(3):187–195.
- LEURGANS, S. and ROSS, R.T. 1992. Multilinear models: applications in spectroscopy. Statistical Science, 7(3):289–310.
- LOPES, J.A. and MENEZES, J.C. 2003. Industrial fermentation end-product modelling with multilinear PLS. Chemometrics and Intelligent Laboratory Systems, 68(1):75–81.
- NOCEDAL, J. and WRIGHT, S.J. 2006. Numerical Optimization. Springer-Verlag, New York, US, 2nd Edition.
- NORELL, B., BURDAKOV, O., ANDERSSON, M. and KNUTSSON, H. 2011. Approximate spectral factorization for design of efficient sub-filter sequences. Technical Report LiTHMAT-R-2011-14, Department of Mathematics, Linköping University.
- ORTEGA, J.M. and RHEINBOLDT, W.C. 2000. Iterative Solution of Nonlinear Equations in Several Variables, Classics in Applied Mathematics 30. SIAM, Philadelphia, USA. Reprint of the 1970 Original Edition.
- PAATERO, P. 1997. Least squares formulation of robust non-negative factor analysis. Chemometrics and Intelligent Laboratory Systems, 37(1):23–35.
- SVENSSON, B., ANDERSSON, M. and KNUTSSON, H. 2005. A graph representation of filter networks. In Proceedings of the 14th Scandinavian Conference on Image Analysis (SCIA’05), pp. 1086–1095, Joensuu, Finland, June 2005.
- WANG, J.H., HOPKE, P.K., HANCEWICZ, T.M. and ZHANG, S.L. 2003. Application of modified alternating least squares regression to spectroscopic image analysis. Analytica Chimica Acta, 476(1):93–109.
References
ANDERSSON, M., WIKLUND, J. and KNUTSSON, H. 1999. Filter networks. In N.M. NAMAZI, editor, Signal and Image Processing (SIP), Proceedings of the IASTED International Conferences, October 18-21, 1999, Nassau, The Bahamas, pages 213–217. IASTED/ACTA Press.
BJÖRK, Å. 1996. Numerical Methods for Least Squares Problems. SIAM, Philadelphia, USA.
GRANLUND, G. and KNUTSSON, H. 1995. Signal Processing for Computer Vision. Kluwer, Dordrecht.
KNUTSSON, H. 1982. Filtering and Reconstruction in Image Processing. PhD thesis, Linköping University, Sweden, Diss. No. 88.
KNUTSSON, H. and ANDERSSON, M. 2005. Implications of invariance and uncertainty for local structure analysis filter sets. Signal Processing: Image Communications, 20(6):569–581.
KNUTSSON, H., WESTIN, C-F. and ANDERSSON, M. 2011. Representing local structure using tensors II. In A. HEYDEN and F. KAHL, Editors, Image Analysis, Lecture Notes in Computer Science, 6688: 545–556. Springer, Berlin/Heidelberg.
LEARDI, R., ARMANINO, C., LANTERI, S. and ALBEROTANZA, L. 2000. Three-mode principal component analysis of monitoring data from Venice lagoon. Journal of Chemometrics, 14(3):187–195.
LEURGANS, S. and ROSS, R.T. 1992. Multilinear models: applications in spectroscopy. Statistical Science, 7(3):289–310.
LOPES, J.A. and MENEZES, J.C. 2003. Industrial fermentation end-product modelling with multilinear PLS. Chemometrics and Intelligent Laboratory Systems, 68(1):75–81.
NOCEDAL, J. and WRIGHT, S.J. 2006. Numerical Optimization. Springer-Verlag, New York, US, 2nd Edition.
NORELL, B., BURDAKOV, O., ANDERSSON, M. and KNUTSSON, H. 2011. Approximate spectral factorization for design of efficient sub-filter sequences. Technical Report LiTHMAT-R-2011-14, Department of Mathematics, Linköping University.
ORTEGA, J.M. and RHEINBOLDT, W.C. 2000. Iterative Solution of Nonlinear Equations in Several Variables, Classics in Applied Mathematics 30. SIAM, Philadelphia, USA. Reprint of the 1970 Original Edition.
PAATERO, P. 1997. Least squares formulation of robust non-negative factor analysis. Chemometrics and Intelligent Laboratory Systems, 37(1):23–35.
SVENSSON, B., ANDERSSON, M. and KNUTSSON, H. 2005. A graph representation of filter networks. In Proceedings of the 14th Scandinavian Conference on Image Analysis (SCIA’05), pp. 1086–1095, Joensuu, Finland, June 2005.
WANG, J.H., HOPKE, P.K., HANCEWICZ, T.M. and ZHANG, S.L. 2003. Application of modified alternating least squares regression to spectroscopic image analysis. Analytica Chimica Acta, 476(1):93–109.