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Multilinear principal component analysis
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<h2 id="the-algorithm">The algorithm</h2> <p>The MPCA solution follows the alternating least square (ALS) approach. It is iterative in nature. As in PCA, MPCA works on centered data. Centering is a little more complicated for tensors, and it is problem dependent. </p> <h2 id="feature-selection">Feature selection</h2> <p>MPCA features: Supervised MPCA is employed in causal factor analysis that facilitates object recognition<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> while a semi-supervised MPCA feature selection is employed in visualization tasks.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> </p> <h2 id="extensions">Extensions</h2> <p>Various extension of MPCA: </p> <ul><li>Robust MPCA (RMPCA)<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a></li> <li>Multi-Tensor Factorization, that also finds the number of components automatically (MTF)<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a></li></ul> <h2 id="external-links">External links</h2> <ul><li><i>Matlab code</i>: <a href="http://www.mathworks.com/matlabcentral/fileexchange/26168">MPCA</a>.</li> <li><i>Matlab code</i>: <a href="http://www.mathworks.com/matlabcentral/fileexchange/35432">UMPCA (including data)</a>.</li> <li><i>R code:</i> <a href="http://research.cs.aalto.fi/pml/software/mtf/">MTF</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>F. L. Hitchcock (1927). "The expression of a tensor or a polyadic as a sum of products". Journal of Mathematics and Physics. 6 (1–4): 164–189. doi:10.1002/sapm192761164. <a href="/wiki/F._L._Hitchcock" target="_blank">/wiki/F._L._Hitchcock</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Tucker, Ledyard R (September 1966). "Some mathematical notes on three-mode factor analysis". Psychometrika. 31 (3): 279–311. doi:10.1007/BF02289464. PMID 5221127. <a href="/wiki/Ledyard_R_Tucker" target="_blank">/wiki/Ledyard_R_Tucker</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>P. M. Kroonenberg and J. de Leeuw, Principal component analysis of three-mode data by means of alternating least squares algorithms, Psychometrika, 45 (1980), pp. 69–97. <a href="https://doi.org/10.1007%2FBF02293599" target="_blank">https://doi.org/10.1007%2FBF02293599</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Lathauwer, L.D.; Moor, B.D.; Vandewalle, J. (2000). "A multilinear singular value decomposition". SIAM Journal on Matrix Analysis and Applications. 21 (4): 1253–1278. doi:10.1137/s0895479896305696. <a href="http://portal.acm.org/citation.cfm?id=354398" target="_blank">http://portal.acm.org/citation.cfm?id=354398</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Lathauwer, L. D.; Moor, B. D.; 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. 21 (4): 1324–1342. doi:10.1137/s0895479898346995. <a href="http://portal.acm.org/citation.cfm?id=354405" target="_blank">http://portal.acm.org/citation.cfm?id=354405</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>M.A.O. Vasilescu (2002) "Human Motion Signatures: Analysis, Synthesis, Recognition," Proceedings of International Conference on Pattern Recognition (ICPR 2002), Vol. 3, Quebec City, Canada, Aug, 2002, 456–460. <a href="http://www.media.mit.edu/~maov/motionsignatures/hms_icpr02_corrected.pdf" target="_blank">http://www.media.mit.edu/~maov/motionsignatures/hms_icpr02_corrected.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>M.A.O. Vasilescu, D. Terzopoulos (2002) "Multilinear Analysis of Image Ensembles: TensorFaces," Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002, in Computer Vision – ECCV 2002, Lecture Notes in Computer Science, Vol. 2350, A. Heyden et al. (Eds.), Springer-Verlag, Berlin, 2002, 447–460. <a href="/wiki/Demetri_Terzopoulos" target="_blank">/wiki/Demetri_Terzopoulos</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>M.A.O. Vasilescu, D. Terzopoulos (2003) "Multilinear Subspace Analysis for Image Ensembles, M. A. O. Vasilescu, D. Terzopoulos, Proc. Computer Vision and Pattern Recognition Conf. (CVPR '03), Vol.2, Madison, WI, June, 2003, 93–99. <a href="http://www.media.mit.edu/~maov/tensorfaces/cvpr03.pdf" target="_blank">http://www.media.mit.edu/~maov/tensorfaces/cvpr03.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>M.A.O. Vasilescu, D. Terzopoulos (2004) "TensorTextures: Multilinear Image-Based Rendering", M. A. O. Vasilescu and D. Terzopoulos, Proc. ACM SIGGRAPH 2004 Conference Los Angeles, CA, August, 2004, in Computer Graphics Proceedings, Annual Conference Series, 2004, 336–342. <a href="http://www.media.mit.edu/~maov/tensortextures/Vasilescu_siggraph04.pdf" target="_blank">http://www.media.mit.edu/~maov/tensortextures/Vasilescu_siggraph04.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>P. M. Kroonenberg and J. de Leeuw, Principal component analysis of three-mode data by means of alternating least squares algorithms, Psychometrika, 45 (1980), pp. 69–97. <a href="https://doi.org/10.1007%2FBF02293599" target="_blank">https://doi.org/10.1007%2FBF02293599</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>M. A. O. Vasilescu, D. Terzopoulos (2005) "Multilinear Independent Component Analysis", "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’05), San Diego, CA, June 2005, vol.1, 547–553." <a href="http://www.media.mit.edu/~maov/mica/mica05.pdf" target="_blank">http://www.media.mit.edu/~maov/mica/mica05.pdf</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>M.A.O. Vasilescu (2002) "Human Motion Signatures: Analysis, Synthesis, Recognition," Proceedings of International Conference on Pattern Recognition (ICPR 2002), Vol. 3, Quebec City, Canada, Aug, 2002, 456–460. <a href="http://www.media.mit.edu/~maov/motionsignatures/hms_icpr02_corrected.pdf" target="_blank">http://www.media.mit.edu/~maov/motionsignatures/hms_icpr02_corrected.pdf</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>M.A.O. Vasilescu, D. Terzopoulos (2002) "Multilinear Analysis of Image Ensembles: TensorFaces," Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002, in Computer Vision – ECCV 2002, Lecture Notes in Computer Science, Vol. 2350, A. Heyden et al. (Eds.), Springer-Verlag, Berlin, 2002, 447–460. <a href="/wiki/Demetri_Terzopoulos" target="_blank">/wiki/Demetri_Terzopoulos</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>M.A.O. Vasilescu, D. Terzopoulos (2003) "Multilinear Subspace Analysis for Image Ensembles, M. A. O. Vasilescu, D. Terzopoulos, Proc. Computer Vision and Pattern Recognition Conf. (CVPR '03), Vol.2, Madison, WI, June, 2003, 93–99. <a href="http://www.media.mit.edu/~maov/tensorfaces/cvpr03.pdf" target="_blank">http://www.media.mit.edu/~maov/tensorfaces/cvpr03.pdf</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>M.A.O. Vasilescu, D. Terzopoulos (2004) "TensorTextures: Multilinear Image-Based Rendering", M. A. O. Vasilescu and D. Terzopoulos, Proc. ACM SIGGRAPH 2004 Conference Los Angeles, CA, August, 2004, in Computer Graphics Proceedings, Annual Conference Series, 2004, 336–342. <a href="http://www.media.mit.edu/~maov/tensortextures/Vasilescu_siggraph04.pdf" target="_blank">http://www.media.mit.edu/~maov/tensortextures/Vasilescu_siggraph04.pdf</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>M. A. O. Vasilescu, D. Terzopoulos (2005) "Multilinear Independent Component Analysis", "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’05), San Diego, CA, June 2005, vol.1, 547–553." <a href="http://www.media.mit.edu/~maov/mica/mica05.pdf" target="_blank">http://www.media.mit.edu/~maov/mica/mica05.pdf</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>M.A.O. Vasilescu (2002) "Human Motion Signatures: Analysis, Synthesis, Recognition," Proceedings of International Conference on Pattern Recognition (ICPR 2002), Vol. 3, Quebec City, Canada, Aug, 2002, 456–460. <a href="http://www.media.mit.edu/~maov/motionsignatures/hms_icpr02_corrected.pdf" target="_blank">http://www.media.mit.edu/~maov/motionsignatures/hms_icpr02_corrected.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>M.A.O. Vasilescu, D. Terzopoulos (2002) "Multilinear Analysis of Image Ensembles: TensorFaces," Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002, in Computer Vision – ECCV 2002, Lecture Notes in Computer Science, Vol. 2350, A. Heyden et al. (Eds.), Springer-Verlag, Berlin, 2002, 447–460. <a href="/wiki/Demetri_Terzopoulos" target="_blank">/wiki/Demetri_Terzopoulos</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>M.A.O. Vasilescu, D. Terzopoulos (2003) "Multilinear Subspace Analysis for Image Ensembles, M. A. O. Vasilescu, D. Terzopoulos, Proc. Computer Vision and Pattern Recognition Conf. (CVPR '03), Vol.2, Madison, WI, June, 2003, 93–99. <a href="http://www.media.mit.edu/~maov/tensorfaces/cvpr03.pdf" target="_blank">http://www.media.mit.edu/~maov/tensorfaces/cvpr03.pdf</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>M.A.O. Vasilescu, D. Terzopoulos (2004) "TensorTextures: Multilinear Image-Based Rendering", M. A. O. Vasilescu and D. Terzopoulos, Proc. ACM SIGGRAPH 2004 Conference Los Angeles, CA, August, 2004, in Computer Graphics Proceedings, Annual Conference Series, 2004, 336–342. <a href="http://www.media.mit.edu/~maov/tensortextures/Vasilescu_siggraph04.pdf" target="_blank">http://www.media.mit.edu/~maov/tensortextures/Vasilescu_siggraph04.pdf</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>M. A. O. Vasilescu, D. Terzopoulos (2003) "Multilinear Subspace Analysis of Image Ensembles", "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’03), Madison, WI, June, 2003" <a href="http://www.cs.toronto.edu/~maov/tensorfaces/cvpr03.pdf" target="_blank">http://www.cs.toronto.edu/~maov/tensorfaces/cvpr03.pdf</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>H. Lu, H.-L. Eng, M. Thida, and K.N. Plataniotis, "Visualization and Clustering of Crowd Video Content in MPCA Subspace," in Proceedings of the 19th ACM Conference on Information and Knowledge Management (CIKM 2010), Toronto, ON, Canada, October, 2010. <a href="http://www.dsp.utoronto.ca/~haiping/Publication/CrowdMPCA_CIKM2010.pdf" target="_blank">http://www.dsp.utoronto.ca/~haiping/Publication/CrowdMPCA_CIKM2010.pdf</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>K. Inoue, K. Hara, K. Urahama, "Robust multilinear principal component analysis", Proc. IEEE Conference on Computer Vision, 2009, pp. 591–597. <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Khan, Suleiman A.; Leppäaho, Eemeli; Kaski, Samuel (2016-06-10). "Bayesian multi-tensor factorization". Machine Learning. 105 (2): 233–253. arXiv:1412.4679. doi:10.1007/s10994-016-5563-y. ISSN 0885-6125. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> </ol>