Content deleted Content added
No edit summary Tag: Reverted |
|||
Line 1:
{{Short description|Tensor decomposition}}
In [[multilinear algebra]], the '''higher-order singular value decomposition''' ('''HOSVD''') of a [[tensor]] is a specific orthogonal [[Tucker decomposition]]. It
<ref name=":2">{{Cite journal|last1=De Lathauwer|first1=L.|last2=De Moor|first2=B.|last3=Vandewalle|first3=J.|date=2000-01-01|title=On the Best Rank-1 and Rank-(R1 ,R2 ,. . .,RN) Approximation of Higher-Order Tensors|journal=SIAM Journal on Matrix Analysis and Applications|volume=21|issue=4|pages=1324–1342|doi=10.1137/S0895479898346995|issn=0895-4798|citeseerx=10.1.1.102.9135}}</ref>▼
The conceptual foundations of tensor decompositions can be traced to [[F. L. Hitchcock]] (1928),<ref name=":0">{{Cite journal|last=Hitchcock|first=Frank L|date=1928-04-01|title=Multiple Invariants and Generalized Rank of a M-Way Array or Tensor|journal=Journal of Mathematics and Physics|language=en|volume=7|issue=1–4|pages=39–79|doi=10.1002/sapm19287139|issn=1467-9590}}</ref> and were significantly developed by [[L. R. Tucker]] in the 1960s, who introduced the general Tucker decomposition for third-order tensors.<ref name=":1">{{Cite journal|last=Tucker|first=Ledyard R.|date=1966-09-01|title=Some mathematical notes on three-mode factor analysis|journal=Psychometrika|volume=31|issue=3|pages=279–311|doi=10.1007/bf02289464}}</ref><ref name="Tucker1963">{{Cite journal|last=Tucker|first=L. R.|date=1963|title=Implications of factor analysis of three-way matrices for measurement of change|journal=In C. W. Harris (Ed.), Problems in Measuring Change. Madison, Wis.: Univ. Wis. Press.|pages=122–137}}</ref><ref name="Tucker1964">{{Cite journal|last=Tucker|first=L. R.|date=1964|title=The extension of factor analysis to three-dimensional matrices|journal=In N. Frederiksen and H. Gulliksen (Eds.), Contributions to Mathematical Psychology. New York: Holt, Rinehart and Winston|pages=109–127}}</ref>
The algorithm widely referred to in the literature as the Tucker or Higher-Order Singular Value Decomposition (HOSVD) was developed by Vasilescu and Terzopoulos, who introduced it under the name M-mode SVD,<ref name=":Vasilescu2002">M. A. O. Vasilescu, D. Terzopoulos (2002), "Multilinear Analysis of Image Ensembles: TensorFaces," Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark. {{Webarchive|url=https://web.archive.org/web/20221229090931/http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf |date=2022-12-29}}</ref><ref name="Vasilescu2003">M. A. O. Vasilescu, D. Terzopoulos (2003), "Multilinear Subspace Analysis of Image Ensembles," Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR’03), Madison, WI.</ref><ref name=":Vasilescu2005">M. A. O. Vasilescu, D. Terzopoulos (2005), "Multilinear Independent Component Analysis," Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR’05), San Diego, CA.</ref> but frequently misattributed to either [[L. R. Tucker]] or [[Lieven De Lathauwer]]. While the term HOSVD was coined by De Lathauwer, it is the M-mode SVD introduced by Vasilescu that is most often—and incorrectly—referred to under that name.▼
▲Later, [[Lieven De Lathauwer]] ''et al.'' formalized the term HOSVD and proposed sequential algorithms based on the power method.<ref name=":2">{{Cite journal|last1=De Lathauwer|first1=L.|last2=De Moor|first2=B.|last3=Vandewalle|first3=J.|date=2000-01-01|title=On the Best Rank-1 and Rank-(R1 ,R2 ,. . .,RN) Approximation of Higher-Order Tensors|journal=SIAM Journal on Matrix Analysis and Applications|volume=21|issue=4|pages=1324–1342|doi=10.1137/S0895479898346995}}</ref><ref name="DeLathauwerSVD">{{Cite journal|
▲The algorithm widely referred to
: This misattribution has had lasting impact on the scholarly record, obscuring the original source of a widely adopted algorithm, and complicating efforts to trace its development, reproduce results, or properly credit foundational contributions in multilinear algebra and tensor methods.
| |||