{"title":"An Improved Algorithm for Calculation of the Third-order Orthogonal Tensor Product Expansion by Using Singular Value Decomposition","authors":"Chiharu Okuma, Naoki Yamamoto, Jun Murakami","volume":38,"journal":"International Journal of Computer and Information Engineering","pagesStart":193,"pagesEnd":203,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/14985","abstract":"
As a method of expanding a higher-order tensor data to tensor products of vectors we have proposed the Third-order Orthogonal Tensor Product Expansion (3OTPE) that did similar expansion as Higher-Order Singular Value Decomposition (HOSVD). In this paper we provide a computation algorithm to improve our previous method, in which SVD is applied to the matrix that constituted by the contraction of original tensor data and one of the expansion vector obtained. The residual of the improved method is smaller than the previous method, truncating the expanding tensor products to the same number of terms. Moreover, the residual is smaller than HOSVD when applying to color image data. It is able to be confirmed that the computing time of improved method is the same as the previous method and considerably better than HOSVD.<\/p>\r\n","references":"[1] Chiharu Okuma, Jun Murakami, and Naoki Yamamoto: Comparison\r\nbetween Higher-order SVD and Third-order Orthogonal Tensor Product\r\nExpansion, International Journal Electronics, Communications and\r\nComputer Engineering, vol.1, no.2, pp.131-137, 2009.\r\n[2] Jun Murakami, Naoki Yamamoto, and Yoshiaki Tadokoro: High-Speed\r\nComputation of 3D Tensor Product Expansion by the Power Method,\r\nElectronics and Communications in Japan, Part 3, Vol.85, pp.63-72,\r\n2002.\r\n[3] Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle: A Multilinear\r\nSingular Value Decomposition, SIAM Journal on Matrix Analysis and\r\nApplications, Vol.21, No.4, pp.1253- 1278, 2000.\r\n[4] Manolis G. Vozalis and Konstantinos G. Margaritis: Applying SVD on\r\nGeneralized Item-based Filtering, International Journal of Computer\r\nScience & Applications, Vol.3, Issue 3, pp.27-51, 2006.\r\n[5] Berkant Savas and Lars Eld\u00e9n: Handwritten Digit Classification using\r\nHigher order Singular Value Decomposition, Pattern Recognition,\r\nVol.40, pp.993-1003, 2007.\r\n[6] J.H. Wilkinson: The Algebraic Eigenvalue Problem, Oxford Science\r\nPublications, 1965.\r\n[7] Gene Howard Golub, Christian Reinsch; Singular Value Decomposition\r\nand Least Squares Solutions, Numerische Mathematik14, pp.403-420,\r\n1970.\r\n[8] Tian bo Deng and Masayuki Kawamata: Design of Two-Dimensional\r\nRecursive Digital Filters Based on the Iterative Singular Value\r\nDecomposition, Transactions of the Institute of Electronics, Information\r\nand Communication Engineers, Vol.E 73, No.6, pp.882-892, 1990.\r\n[9] Makoto Ohki and Masayuki Kawamata: Design of Three-Dimensional\r\nDigital Filters Based on the Outer Product Expansion, IEEE Transactions\r\non circuits and Systems, Vol.CAS-37, No.9, pp.1164-1167, 1990.\r\n[10] R. L. Johnston: Numerical Methods, John Wiley & Sons, 1982.\r\n[11] Jamie Hutchinson: Culture Communication, and an Information Age\r\nMadonna IEEE Professional Communication Society Newsletter,\r\nVolume 45, No 3, pp. 1, 5-7, 2001.\r\n[12] Taro Konda, Masami Takata, Masashi Iwasaki, and Yoshimasa\r\nNakamura: A new singular value decomposition algorithm suited to\r\nparallelization and preliminary results, Proceedings of the 2nd IASTED\r\ninternational conference on Advances in computer science and\r\ntechnology, Puerto Vallarta, Mexico, pp.79-84, 2006.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 38, 2010"}