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On the Best Rank-1 and Rank-(R1 ,R2 ,. . .,RN) Approximation of Higher-Order Tensors
Abstract
In this paper we discuss a multilinear generalization of the best rank-R approximation problem for matrices, namely, the approximation of a given higher-order tensor, in an optimal least-squares sense, by a tensor that has prespecified column rank value, row rank value, etc. For matrices, the solution is conceptually obtained by truncation of the singular value decomposition (SVD); however, this approach does not have a straightforward multilinear counterpart. We discuss higher-order generalizations of the power method and the orthogonal iteration method.
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Cited by
Ghamisi, P., Yokoya, N., Li, J., Liao, W., Liu, S., Plaza, J., Rasti, B., & Plaza, A. (2017) · IEEE Geoscience and Remote Sensing Magazine · doi:10.1109/mgrs.2017.2762087 · Cited by 909
Multilinear Analysis of Image Ensembles: TensorFaces
Vasilescu, M. A. O., & Terzopoulos, D. (2002) · Lecture Notes in Computer Science · doi:10.1007/3-540-47969-4_30 · Cited by 836
Tensor decompositions for learning latent variable models
(2012) · arXiv (Cornell University) · doi:10.5555/2627435.2697055 · Cited by 779
Incremental tensor analysis: Theory and applications
Sun, J., Tao, D., Papadimitriou, S., Yu, P. S., & Faloutsos, C. (2008) · ACM Transactions on Knowledge Discovery from Data · doi:10.1145/1409620.1409621 · Cited by 254
Approximation of $2^d\times2^d$ Matrices Using Tensor Decomposition
Oseledets, I. V. (2010) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/090757861 · Cited by 174
Metadata
- Authors
- De Lathauwer, L., De Moor, B., & Vandewalle, J.
- Venue
- SIAM Journal on Matrix Analysis and Applications
- Published
- 2000-01
- Volume
- 21
- Issue
- 4
- Pages
- 1324-1342
- ISSN
- 0895-4798 (ISSN-L)
- 1095-7162
- OpenAlex
- W2018282388
- Cited by
- 1,567
- References
- 0
- FWCI
- 2.24
- Type
- journal article
- Language
- en
- Open access
- closed
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- De Lathauwer, L., De Moor, B., & Vandewalle, J. (2000). On the Best Rank-1 and Rank-(R1 ,R2 ,. . .,RN) Approximation of Higher-Order Tensors. SIAM Journal on Matrix Analysis and Applications. https://doi.org/10.1137/s0895479898346995
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