An Improved Algorithm for Calculation of the Third-order Orthogonal Tensor Product Expansion by Using Singular Value Decomposition
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.
[1] Chiharu Okuma, Jun Murakami, and Naoki Yamamoto: Comparison
between Higher-order SVD and Third-order Orthogonal Tensor Product
Expansion, International Journal Electronics, Communications and
Computer Engineering, vol.1, no.2, pp.131-137, 2009.
[2] Jun Murakami, Naoki Yamamoto, and Yoshiaki Tadokoro: High-Speed
Computation of 3D Tensor Product Expansion by the Power Method,
Electronics and Communications in Japan, Part 3, Vol.85, pp.63-72,
2002.
[3] Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle: A Multilinear
Singular Value Decomposition, SIAM Journal on Matrix Analysis and
Applications, Vol.21, No.4, pp.1253- 1278, 2000.
[4] Manolis G. Vozalis and Konstantinos G. Margaritis: Applying SVD on
Generalized Item-based Filtering, International Journal of Computer
Science & Applications, Vol.3, Issue 3, pp.27-51, 2006.
[5] Berkant Savas and Lars Eldén: Handwritten Digit Classification using
Higher order Singular Value Decomposition, Pattern Recognition,
Vol.40, pp.993-1003, 2007.
[6] J.H. Wilkinson: The Algebraic Eigenvalue Problem, Oxford Science
Publications, 1965.
[7] Gene Howard Golub, Christian Reinsch; Singular Value Decomposition
and Least Squares Solutions, Numerische Mathematik14, pp.403-420,
1970.
[8] Tian bo Deng and Masayuki Kawamata: Design of Two-Dimensional
Recursive Digital Filters Based on the Iterative Singular Value
Decomposition, Transactions of the Institute of Electronics, Information
and Communication Engineers, Vol.E 73, No.6, pp.882-892, 1990.
[9] Makoto Ohki and Masayuki Kawamata: Design of Three-Dimensional
Digital Filters Based on the Outer Product Expansion, IEEE Transactions
on circuits and Systems, Vol.CAS-37, No.9, pp.1164-1167, 1990.
[10] R. L. Johnston: Numerical Methods, John Wiley & Sons, 1982.
[11] Jamie Hutchinson: Culture Communication, and an Information Age
Madonna IEEE Professional Communication Society Newsletter,
Volume 45, No 3, pp. 1, 5-7, 2001.
[12] Taro Konda, Masami Takata, Masashi Iwasaki, and Yoshimasa
Nakamura: A new singular value decomposition algorithm suited to
parallelization and preliminary results, Proceedings of the 2nd IASTED
international conference on Advances in computer science and
technology, Puerto Vallarta, Mexico, pp.79-84, 2006.
[1] Chiharu Okuma, Jun Murakami, and Naoki Yamamoto: Comparison
between Higher-order SVD and Third-order Orthogonal Tensor Product
Expansion, International Journal Electronics, Communications and
Computer Engineering, vol.1, no.2, pp.131-137, 2009.
[2] Jun Murakami, Naoki Yamamoto, and Yoshiaki Tadokoro: High-Speed
Computation of 3D Tensor Product Expansion by the Power Method,
Electronics and Communications in Japan, Part 3, Vol.85, pp.63-72,
2002.
[3] Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle: A Multilinear
Singular Value Decomposition, SIAM Journal on Matrix Analysis and
Applications, Vol.21, No.4, pp.1253- 1278, 2000.
[4] Manolis G. Vozalis and Konstantinos G. Margaritis: Applying SVD on
Generalized Item-based Filtering, International Journal of Computer
Science & Applications, Vol.3, Issue 3, pp.27-51, 2006.
[5] Berkant Savas and Lars Eldén: Handwritten Digit Classification using
Higher order Singular Value Decomposition, Pattern Recognition,
Vol.40, pp.993-1003, 2007.
[6] J.H. Wilkinson: The Algebraic Eigenvalue Problem, Oxford Science
Publications, 1965.
[7] Gene Howard Golub, Christian Reinsch; Singular Value Decomposition
and Least Squares Solutions, Numerische Mathematik14, pp.403-420,
1970.
[8] Tian bo Deng and Masayuki Kawamata: Design of Two-Dimensional
Recursive Digital Filters Based on the Iterative Singular Value
Decomposition, Transactions of the Institute of Electronics, Information
and Communication Engineers, Vol.E 73, No.6, pp.882-892, 1990.
[9] Makoto Ohki and Masayuki Kawamata: Design of Three-Dimensional
Digital Filters Based on the Outer Product Expansion, IEEE Transactions
on circuits and Systems, Vol.CAS-37, No.9, pp.1164-1167, 1990.
[10] R. L. Johnston: Numerical Methods, John Wiley & Sons, 1982.
[11] Jamie Hutchinson: Culture Communication, and an Information Age
Madonna IEEE Professional Communication Society Newsletter,
Volume 45, No 3, pp. 1, 5-7, 2001.
[12] Taro Konda, Masami Takata, Masashi Iwasaki, and Yoshimasa
Nakamura: A new singular value decomposition algorithm suited to
parallelization and preliminary results, Proceedings of the 2nd IASTED
international conference on Advances in computer science and
technology, Puerto Vallarta, Mexico, pp.79-84, 2006.
@article{"International Journal of Information, Control and Computer Sciences:63984", author = "Chiharu Okuma and Naoki Yamamoto and Jun Murakami", title = "An Improved Algorithm for Calculation of the Third-order Orthogonal Tensor Product Expansion by Using Singular Value Decomposition", 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.
", keywords = "Singular value decomposition (SVD), higher-orderSVD (HOSVD), outer product expansion, power method.", volume = "4", number = "2", pages = "344-10", }
