Content deleted Content added
No edit summary |
No edit summary |
||
Line 21:
}}</ref> and by the "3-mode PCA" by Kroonenberg<ref name="Kroonenberg1980">P. M. Kroonenberg and J. de Leeuw, [https://doi.org/10.1007%2FBF02293599 Principal component analysis of three-mode data by means of alternating least squares algorithms], Psychometrika, 45 (1980), pp. 69–97</ref> Kroonenbeg's algorithm is an itterative algorithm that employs gradient descent. In 2000, De Lathauwer et al. restated Tucker and Kroonenberg's work in clear terms in their SIAM paper entitled "[[Multilinear Singular Value Decomposition]]",<ref name="DeLathauwer2000a">{{cite journal | last1 = Lathauwer | first1 = L.D. | last2 = Moor | first2 = B.D. | last3 = Vandewalle | first3 = J. | year = 2000 | title = A multilinear singular value decomposition | url = http://portal.acm.org/citation.cfm?id=354398 | journal = SIAM Journal on Matrix Analysis and Applications | volume = 21 | issue = 4| pages = 1253–1278 | doi = 10.1137/s0895479896305696 | url-access = subscription }}</ref> and provided an itterative algorithm that employed the power method in their paper "On the Best Rank-1 and Rank-(R<sub>1</sub>, R<sub>2</sub>, ..., R<sub>N</sub> ) Approximation of Higher-order Tensors".<ref name=DeLathauwer2000b>{{cite journal | last1 = Lathauwer | first1 = L. D. | last2 = Moor | first2 = B. D. | last3 = Vandewalle | first3 = J. | year = 2000 | title = On the best rank-1 and rank-(R1, R2, ..., RN ) approximation of higher-order tensors | url = http://portal.acm.org/citation.cfm?id=354405 | journal = SIAM Journal on Matrix Analysis and Applications | volume = 21 | issue = 4| pages = 1324–1342 | doi = 10.1137/s0895479898346995 | url-access = subscription }}</ref>
Vasilescu and Terzopoulos in their paper "[[Multilinear Image Representation: TensorFaces]]"<ref name=Vasilescu2002a/> introduced the [[HOSVD| M-mode SVD]] algorithm which is a simple and elegant algorithm suitable for parallel computation. This algorithm is often misidentified in the literature as the HOSVD or the Tucker which are sequential itterative algorithms that employ gradient descent. Vasilescu and Terzopoulos framed the data analysis, recognition and synthesis problems as multilinear tensor problems. Data is viewed as the compositional consequence of several causal factors, and which are well suited for multi-modal tensor factor analysis. The power of the tensor framework was showcased by analyzing human motion joint angles, facial images or textures in the following papers: Human Motion Signatures<ref name="Vasilescu2002b">M.A.O. Vasilescu (2002) [http://www.media.mit.edu/~maov/motionsignatures/hms_icpr02_corrected.pdf "Human Motion Signatures: Analysis, Synthesis, Recognition," Proceedings of International Conference on Pattern Recognition (ICPR 2002), Vol. 3, Quebec City, Canada, Aug, 2002, 456–460.]</ref>
(CVPR 2001, ICPR 2002), face recognition – [[TensorFaces]],<ref name="Vasilescu2002a"/><ref name="Vasilescu2003"/>
(ECCV 2002, CVPR 2003, etc.) and computer graphics – [[TensorTextures]]<ref name="Vasilescu2004"/> (Siggraph 2004).
| |||