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Multilinear subspace learning
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<h2 id="background">Background</h2> <p>Multilinear methods may be causal in nature and perform causal inference, or they may be simple regression methods from which no causal conclusion are drawn. </p><p><a href="/facts/Linear_subspace/2wtKTgHL">Linear subspace</a> learning algorithms are traditional dimensionality reduction techniques that are well suited for datasets that are the result of varying a single causal factor. Unfortunately, they often become inadequate when dealing with datasets that are the result of multiple causal factors. . </p><p>Multilinear subspace learning can be applied to observations whose measurements were vectorized and organized into a data tensor for causally aware dimensionality reduction.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> These methods may also be employed in reducing horizontal and vertical redundancies irrespective of the causal factors when the observations are treated as a "matrix" (ie. a collection of independent column/row observations) and concatenated into a tensor.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p> <h2 id="algorithms">Algorithms</h2> <h3>Multilinear principal component analysis</h3> <p>Historically, <a href="/facts/Multilinear_principal_component_analysis/HDRc9a8J">multilinear principal component analysis</a> has been referred to as "M-mode PCA", a terminology which was coined by Peter Kroonenberg.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> In 2005, Vasilescu and <a href="/facts/Demetri_Terzopoulos/S11NpKYM">Terzopoulos</a> introduced the Multilinear PCA<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> terminology as a way to better differentiate between multilinear tensor decompositions that computed 2nd order statistics associated with each data tensor mode,<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a><a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> and subsequent work on Multilinear Independent Component Analysis<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> that computed higher order statistics for each tensor mode. MPCA is an extension of <a href="/facts/Principal_component_analysis/pCbVTqdR">PCA</a>. </p> <h3>Multilinear independent component analysis</h3> <p>Multilinear independent component analysis<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> is an extension of <a href="/facts/Independent_component_analysis/gsdVaDFR">ICA</a>. </p> <h3>Multilinear linear discriminant analysis</h3> <ul><li>Multilinear extension of <a href="/facts/Linear_discriminant_analysis/Mj6KQhRN">LDA</a> <ul><li>TTP-based: Discriminant Analysis with Tensor Representation (DATER)<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a></li> <li>TTP-based: General tensor discriminant analysis (GTDA)<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a></li> <li>TVP-based: Uncorrelated Multilinear Discriminant Analysis (UMLDA)<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a></li></ul></li></ul> <h3>Multilinear canonical correlation analysis</h3> <ul><li>Multilinear extension of <a href="/facts/Canonical_correlation_analysis/LuX8FltG">CCA</a> <ul><li>TTP-based: Tensor Canonical Correlation Analysis (TCCA)<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a></li> <li>TVP-based: Multilinear Canonical Correlation Analysis (MCCA)<a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a></li> <li>TVP-based: Bayesian Multilinear Canonical Correlation Analysis (BMTF)<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a></li></ul></li> <li>A TTP is a direct projection of a high-dimensional tensor to a low-dimensional tensor of the same order, using <i>N</i> projection matrices for an <i>N</i>th-order tensor. It can be performed in <i>N</i> steps with each step performing a tensor-matrix multiplication (product). The <i>N</i> steps are exchangeable.<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> This projection is an extension of the <a href="/facts/Higher-order_singular_value_decomposition/JTtTDy39">higher-order singular value decomposition</a><a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a> (HOSVD) to subspace learning.<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a> Hence, its origin is traced back to the <a href="/facts/Tucker_decomposition/qkef5Dst">Tucker decomposition</a><a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a> in 1960s.</li></ul> <ul><li>A TVP is a direct projection of a high-dimensional tensor to a low-dimensional vector, which is also referred to as the rank-one projections. As TVP projects a tensor to a vector, it can be viewed as multiple projections from a tensor to a scalar. Thus, the TVP of a tensor to a <i>P</i>-dimensional vector consists of <i>P</i> projections from the tensor to a scalar. The projection from a tensor to a scalar is an elementary multilinear projection (EMP). In EMP, a tensor is projected to a point through <i>N</i> unit projection vectors. It is the projection of a tensor on a single line (resulting a scalar), with one projection vector in each mode. Thus, the TVP of a tensor object to a vector in a <i>P</i>-dimensional vector space consists of <i>P</i> EMPs. This projection is an extension of the <a href="/facts/CP_decomposition/sL3JMvYx">canonical decomposition</a>,<a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a> also known as the <a href="/facts/PARAFAC/sL3JMvYx">parallel factors</a> (PARAFAC) decomposition.<a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a></li></ul> <h3>Typical approach in MSL</h3> <p>There are <i>N</i> sets of parameters to be solved, one in each mode. The solution to one set often depends on the other sets (except when <i>N=1</i>, the linear case). Therefore, the suboptimal iterative procedure in<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a> is followed. </p> <ol><li>Initialization of the projections in each mode</li> <li>For each mode, fixing the projection in all the other mode, and solve for the projection in the current mode.</li> <li>Do the mode-wise optimization for a few iterations or until convergence.</li></ol> <p>This is originated from the alternating least square method for multi-way data analysis.<a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a> </p> <h2 id="code">Code</h2> <ul><li><a href="https://web.archive.org/web/20110717172720/http://csmr.ca.sandia.gov/~tgkolda/TensorToolbox/">MATLAB Tensor Toolbox</a> by <a href="/facts/Sandia_National_Laboratories/LYsKDj45">Sandia National Laboratories</a>.</li> <li><a href="http://www.mathworks.com/matlabcentral/fileexchange/26168">The MPCA algorithm written in Matlab (MPCA+LDA included)</a>.</li> <li><a href="http://www.mathworks.com/matlabcentral/fileexchange/35432">The UMPCA algorithm written in Matlab (data included)</a>.</li> <li><a href="http://www.mathworks.fr/matlabcentral/fileexchange/35782">The UMLDA algorithm written in Matlab (data included)</a>.</li></ul> <h2 id="tensor-data-sets">Tensor data sets</h2> <ul><li>3D gait data (third-order tensors): <a href="http://www.dsp.utoronto.ca/~haiping/CodeData/USFGait17_128x88x20.zip">128x88x20(21.2M)</a>; <a href="http://www.dsp.utoronto.ca/~haiping/CodeData/USFGait17_64x44x20.zip">64x44x20(9.9M)</a>; <a href="http://www.dsp.utoronto.ca/~haiping/CodeData/USFGait17_32x22x10.zip">32x22x10(3.2M)</a>;</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/CP_decomposition/sL3JMvYx">CP decomposition</a></li> <li><a href="/facts/Dimension_reduction/XoRpcrB0">Dimension reduction</a></li> <li><a href="/facts/Multilinear_algebra/LrwpUFsM">Multilinear algebra</a></li> <li><a href="/facts/Multilinear_PCA/HDRc9a8J">Multilinear Principal Component Analysis</a></li> <li><a href="/facts/Tensor/pNjFo5tV">Tensor</a></li> <li><a href="/facts/Tensor_decomposition/5YNsQwfT">Tensor decomposition</a></li> <li><a href="/facts/Tensor_software/S2HWT14o">Tensor software</a></li> <li><a href="/facts/Tucker_decomposition/qkef5Dst">Tucker decomposition</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><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:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><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 <a href="http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf" target="_blank">http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><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:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Vasilescu, M.A.O.; Terzopoulos, D. (2007). Multilinear Projection for Appearance-Based Recognition in the Tensor Framework. IEEE 11th International Conference on Computer Vision. pp. 1–8. doi:10.1109/ICCV.2007.4409067.. <a href="/wiki/International_Conference_on_Computer_Vision" target="_blank">/wiki/International_Conference_on_Computer_Vision</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Lu, Haiping; Plataniotis, K.N.; Venetsanopoulos, A.N. (2013). Multilinear Subspace Learning: Dimensionality Reduction of Multidimensional Data. Chapman & Hall/CRC Press Machine Learning and Pattern Recognition Series. Taylor and Francis. ISBN 978-1-4398572-4-3. <a href="978-1-4398572-4-3" target="_blank">978-1-4398572-4-3</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><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:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Lu, Haiping; Plataniotis, K.N.; Venetsanopoulos, A.N. (2011). "A Survey of Multilinear Subspace Learning for Tensor Data" (PDF). Pattern Recognition. 44 (7): 1540–1551. Bibcode:2011PatRe..44.1540L. doi:10.1016/j.patcog.2011.01.004. <a href="http://www.dsp.utoronto.ca/~haiping/Publication/SurveyMSL_PR2011.pdf" target="_blank">http://www.dsp.utoronto.ca/~haiping/Publication/SurveyMSL_PR2011.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>X. He, D. Cai, P. Niyogi, Tensor subspace analysis, in: Advances in Neural Information Processing Systemsc 18 (NIPS), 2005. <a href="http://books.nips.cc/papers/files/nips18/NIPS2005_0249.pdf" target="_blank">http://books.nips.cc/papers/files/nips18/NIPS2005_0249.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Vasilescu, M.A.O.; Terzopoulos, D. (2007). Multilinear Projection for Appearance-Based Recognition in the Tensor Framework. IEEE 11th International Conference on Computer Vision. pp. 1–8. doi:10.1109/ICCV.2007.4409067.. <a href="/wiki/International_Conference_on_Computer_Vision" target="_blank">/wiki/International_Conference_on_Computer_Vision</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Lu, Haiping; Plataniotis, K.N.; Venetsanopoulos, A.N. (2011). "A Survey of Multilinear Subspace Learning for Tensor Data" (PDF). Pattern Recognition. 44 (7): 1540–1551. Bibcode:2011PatRe..44.1540L. doi:10.1016/j.patcog.2011.01.004. <a href="http://www.dsp.utoronto.ca/~haiping/Publication/SurveyMSL_PR2011.pdf" target="_blank">http://www.dsp.utoronto.ca/~haiping/Publication/SurveyMSL_PR2011.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><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:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>"Future Directions in Tensor-Based Computation and Modeling" (PDF). May 2009. <a href="http://www.cs.cornell.edu/cv/tenwork/finalreport.pdf" target="_blank">http://www.cs.cornell.edu/cv/tenwork/finalreport.pdf</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>S. Yan, D. Xu, Q. Yang, L. Zhang, X. Tang, and H.-J. Zhang, "Discriminant analysis with tensor representation," in Proc. IEEE Conference on Computer Vision and Pattern Recognition, vol. I, June 2005, pp. 526–532. <a href="http://portal.acm.org/citation.cfm?id=1068959" target="_blank">http://portal.acm.org/citation.cfm?id=1068959</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><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:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><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:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><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:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><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 <a href="http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf" target="_blank">http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><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:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><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:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, "MPCA: Multilinear principal component analysis of tensor objects," IEEE Trans. Neural Netw., vol. 19, no. 1, pp. 18–39, January 2008. <a href="https://dx.doi.org/10.1109/TNN.2007.901277" target="_blank">https://dx.doi.org/10.1109/TNN.2007.901277</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><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:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><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:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>S. Yan, D. Xu, Q. Yang, L. Zhang, X. Tang, and H.-J. Zhang, "Discriminant analysis with tensor representation," in Proc. IEEE Conference on Computer Vision and Pattern Recognition, vol. I, June 2005, pp. 526–532. <a href="http://portal.acm.org/citation.cfm?id=1068959" target="_blank">http://portal.acm.org/citation.cfm?id=1068959</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>D. Tao, X. Li, X. Wu, and S. J. Maybank, "General tensor discriminant analysis and gabor features for gait recognition," IEEE Trans. Pattern Anal. Mach. Intell., vol. 29, no. 10, pp. 1700–1715, October 2007. <a href="https://dx.doi.org/10.1109/TPAMI.2007.1096" target="_blank">https://dx.doi.org/10.1109/TPAMI.2007.1096</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, "Uncorrelated multilinear discriminant analysis with regularization and aggregation for tensor object recognition," IEEE Trans. Neural Netw., vol. 20, no. 1, pp. 103–123, January 2009. <a href="https://dx.doi.org/10.1109/TNN.2008.2004625" target="_blank">https://dx.doi.org/10.1109/TNN.2008.2004625</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>T.-K. Kim and R. Cipolla. "Canonical correlation analysis of video volume tensors for action categorization and detection," IEEE Trans. Pattern Anal. Mach. Intell., vol. 31, no. 8, pp. 1415–1428, 2009. <a href="https://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4547427" target="_blank">https://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4547427</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>H. Lu, "Learning Canonical Correlations of Paired Tensor Sets via Tensor-to-Vector Projection," Proceedings of the 23rd International Joint Conference on Artificial Intelligence (IJCAI 2013), Beijing, China, August 3–9, 2013. <a href="http://www.dsp.utoronto.ca/~haiping/Publication/MCCA_IJCAI2013.pdf" target="_blank">http://www.dsp.utoronto.ca/~haiping/Publication/MCCA_IJCAI2013.pdf</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>Khan, Suleiman A.; Kaski, Samuel (2014-09-15). "Bayesian Multi-view Tensor Factorization". In Calders, Toon; Esposito, Floriana; Hüllermeier, Eyke; Meo, Rosa (eds.). Machine Learning and Knowledge Discovery in Databases. Lecture Notes in Computer Science. Vol. 8724. Springer Berlin Heidelberg. pp. 656–671. doi:10.1007/978-3-662-44848-9_42. ISBN 9783662448472. <a href="9783662448472" target="_blank">9783662448472</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>L.D. Lathauwer, B.D. Moor, J. 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