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Multilinear subspace learning

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Multilinear subspace learning is an approach for disentangling the causal factor of data formation and performing dimensionality reduction.<a href="#cite_note-Vasilescu2003-1">[1]</a><a href="#cite_note-Vasilescu2002tensorfaces-2">[2]</a><a href="#cite_note-Vasilescu2002hms-3">[3]</a><a href="#cite_note-Vasilescu2007-4">[4]</a><a href="#cite_note-MSLbook-5">[5]</a>

The <a href="Dimension_reduction" class="mw-redirect" title="Dimension reduction">Dimensionality reduction</a> can be performed on a data <a href="Tensor" title="Tensor">tensor</a> that contains a collection of observations that have been vectorized,<a href="#cite_note-Vasilescu2003-1">[1]</a> or observations that are treated as matrices and concatenated into a data tensor.<a href="#cite_note-MSLsurvey-6">[6]</a><a href="#cite_note-TSAnips-7">[7]</a> Here are some examples of data tensors whose observations are vectorized or whose observations are matrices concatenated into data tensor <a href="Image" title="Image">images</a> (2D/3D), <a href="Video" title="Video">video</a> sequences (3D/4D), and <a href="Hyperspectral_imaging" title="Hyperspectral imaging">hyperspectral cubes</a> (3D/4D).

The mapping from a <a href="High-dimensional_vector_space" class="mw-redirect" title="High-dimensional vector space">high-dimensional vector space</a> to a set of lower dimensional <a href="Vector_space" title="Vector space">vector spaces</a> is a multilinear projection.<a href="#cite_note-Vasilescu2007-4">[4]</a> When observations are retained in the same organizational structure as matrices or higher order tensors, their representations are computed by performing linear projections into the column space, row space and fiber space.<a href="#cite_note-MSLsurvey-6">[6]</a>

<a href="#Algorithms">Multilinear subspace learning algorithms</a> are higher-order generalizations of <a href="Linear_subspace" title="Linear subspace">linear subspace</a> learning methods such as <a href="Principal_component_analysis" title="Principal component analysis">principal component analysis</a> (PCA), <a href="Independent_component_analysis" title="Independent component analysis">independent component analysis</a> (ICA), <a href="Linear_discriminant_analysis" title="Linear discriminant analysis">linear discriminant analysis</a> (LDA) and <a href="Canonical_correlation" title="Canonical correlation">canonical correlation analysis</a> (CCA).

<div class="mw-heading mw-heading2"><h2 id="Background">Background</h2></div>

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.

<a href="Linear_subspace" title="Linear subspace">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. .

Multilinear subspace learning can be applied to observations whose measurements were vectorized and organized into a data tensor for causally aware dimensionality reduction.<a href="#cite_note-Vasilescu2003-1">[1]</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 href="#cite_note-8">[8]</a><a href="#cite_note-DATER-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="Algorithms">Algorithms</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Multilinear_principal_component_analysis">Multilinear principal component analysis</h3></div>

Historically, <a href="Multilinear_principal_component_analysis" title="Multilinear principal component analysis">multilinear principal component analysis</a> has been referred to as "M-mode PCA", a terminology which was coined by Peter Kroonenberg.<a href="#cite_note-Kroonenberg1980-10">[10]</a>

In 2005, Vasilescu and <a href="Demetri_Terzopoulos" title="Demetri Terzopoulos">Terzopoulos</a> introduced the Multilinear PCA<a href="#cite_note-MPCA-MICA2005-11">[11]</a> terminology as a way to better differentiate between multilinear tensor decompositions that computed 2nd order statistics associated with each data tensor mode,<a href="#cite_note-Vasilescu2003-1">[1]</a><a href="#cite_note-Vasilescu2002tensorfaces-2">[2]</a><a href="#cite_note-Vasilescu2002hms-3">[3]</a><a href="#cite_note-Vasilescu2004-12">[12]</a><a href="#cite_note-MPCA-Lu2008-13">[13]</a> and subsequent work on Multilinear Independent Component Analysis<a href="#cite_note-MPCA-MICA2005-11">[11]</a> that computed higher order statistics for each tensor mode. MPCA is an extension of <a href="Principal_component_analysis" title="Principal component analysis">PCA</a>.

<div class="mw-heading mw-heading3"><h3 id="Multilinear_independent_component_analysis">Multilinear independent component analysis</h3></div>

Multilinear independent component analysis<a href="#cite_note-MPCA-MICA2005-11">[11]</a> is an extension of <a href="Independent_component_analysis" title="Independent component analysis">ICA</a>.

<div class="mw-heading mw-heading3"><h3 id="Multilinear_linear_discriminant_analysis">Multilinear linear discriminant analysis</h3></div>

<ul><li>Multilinear extension of <a href="Linear_discriminant_analysis" title="Linear discriminant analysis">LDA</a>

<ul><li>TTP-based: Discriminant Analysis with Tensor Representation (DATER)<a href="#cite_note-DATER-9">[9]</a></li>

<li>TTP-based: General tensor discriminant analysis (GTDA)<a href="#cite_note-14">[14]</a></li>

<li>TVP-based: Uncorrelated Multilinear Discriminant Analysis (UMLDA)<a href="#cite_note-15">[15]</a></li></ul></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Multilinear_canonical_correlation_analysis">Multilinear canonical correlation analysis</h3></div>

<ul><li>Multilinear extension of <a href="Canonical_correlation_analysis" class="mw-redirect" title="Canonical correlation analysis">CCA</a>

<ul><li>TTP-based: Tensor Canonical Correlation Analysis (TCCA)<a href="#cite_note-16">[16]</a></li>

<li>TVP-based: Multilinear Canonical Correlation Analysis (MCCA)<a href="#cite_note-17">[17]</a></li>

<li>TVP-based: Bayesian Multilinear Canonical Correlation Analysis (BMTF)<a href="#cite_note-18">[18]</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 N projection matrices for an Nth-order tensor. It can be performed in N steps with each step performing a tensor-matrix multiplication (product). The N steps are exchangeable.<a href="#cite_note-HOSVD-19">[19]</a> This projection is an extension of the <a href="Higher-order_singular_value_decomposition" title="Higher-order singular value decomposition">higher-order singular value decomposition</a><a href="#cite_note-HOSVD-19">[19]</a> (HOSVD) to subspace learning.<a href="#cite_note-MPCA-Lu2008-13">[13]</a> Hence, its origin is traced back to the <a href="Tucker_decomposition" title="Tucker decomposition">Tucker decomposition</a><a href="#cite_note-20">[20]</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 P-dimensional vector consists of P 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 N 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 P-dimensional vector space consists of P EMPs. This projection is an extension of the <a href="CP_decomposition" class="mw-redirect" title="CP decomposition">canonical decomposition</a>,<a href="#cite_note-21">[21]</a> also known as the <a href="PARAFAC" class="mw-redirect" title="PARAFAC">parallel factors</a> (PARAFAC) decomposition.<a href="#cite_note-22">[22]</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Typical_approach_in_MSL">Typical approach in MSL</h3></div>

There are N sets of parameters to be solved, one in each mode. The solution to one set often depends on the other sets (except when N=1, the linear case). Therefore, the suboptimal iterative procedure in<a href="#cite_note-23">[23]</a> is followed.

<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>

This is originated from the alternating least square method for multi-way data analysis.<a href="#cite_note-Kroonenberg1980-10">[10]</a>

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110717172720/http://csmr.ca.sandia.gov/~tgkolda/TensorToolbox/">MATLAB Tensor Toolbox</a> by <a href="Sandia_National_Laboratories" title="Sandia National Laboratories">Sandia National Laboratories</a>.</li>

<li><a rel="nofollow" class="external text" href="http://www.mathworks.com/matlabcentral/fileexchange/26168">The MPCA algorithm written in Matlab (MPCA+LDA included)</a>.</li>

<li><a rel="nofollow" class="external text" href="http://www.mathworks.com/matlabcentral/fileexchange/35432">The UMPCA algorithm written in Matlab (data included)</a>.</li>

<li><a rel="nofollow" class="external text" href="http://www.mathworks.fr/matlabcentral/fileexchange/35782">The UMLDA algorithm written in Matlab (data included)</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Tensor_data_sets">Tensor data sets</h2></div>

<ul><li>3D gait data (third-order tensors): <a rel="nofollow" class="external text" href="http://www.dsp.utoronto.ca/~haiping/CodeData/USFGait17_128x88x20.zip">128x88x20(21.2M)</a>; <a rel="nofollow" class="external text" href="http://www.dsp.utoronto.ca/~haiping/CodeData/USFGait17_64x44x20.zip">64x44x20(9.9M)</a>; <a rel="nofollow" class="external text" href="http://www.dsp.utoronto.ca/~haiping/CodeData/USFGait17_32x22x10.zip">32x22x10(3.2M)</a>;</li></ul>

<ul><li><a href="CP_decomposition" class="mw-redirect" title="CP decomposition">CP decomposition</a></li>

<li><a href="Dimension_reduction" class="mw-redirect" title="Dimension reduction">Dimension reduction</a></li>

<li><a href="Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li>

<li><a href="Multilinear_PCA" class="mw-redirect" title="Multilinear PCA">Multilinear Principal Component Analysis</a></li>

<li><a href="Tensor" title="Tensor">Tensor</a></li>

<li><a href="Tensor_decomposition" title="Tensor decomposition">Tensor decomposition</a></li>

<li><a href="Tensor_software" title="Tensor software">Tensor software</a></li>

<li><a href="Tucker_decomposition" title="Tucker decomposition">Tucker decomposition</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-20"><a href="#cite_ref-20">^</a> <cite id="CITEREFLedyard_R_Tucker1966" class="citation journal cs1"><a href="Ledyard_R_Tucker" class="mw-redirect" title="Ledyard R Tucker">Ledyard R Tucker</a> (September 1966). "Some mathematical notes on three-mode factor analysis". <a href="Psychometrika" title="Psychometrika">Psychometrika</a>. 31 (3): 279–311. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02289464">10.1007/BF02289464</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/5221127">5221127</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:44301099">44301099</a>.</cite>

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<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFJ._D._Carroll_&_J._Chang1970" class="citation journal cs1">J. D. Carroll & J. Chang (1970). "Analysis of individual differences in multidimensional scaling via an n-way generalization of 'Eckart–Young' decomposition". <a href="Psychometrika" title="Psychometrika">Psychometrika</a>. 35 (3): 283–319. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02310791">10.1007/BF02310791</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:50364581">50364581</a>.</cite>

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<li id="cite_note-22"><a href="#cite_ref-22">^</a> R. A. Harshman, <a rel="nofollow" class="external text" href="http://publish.uwo.ca/~harshman/wpppfac0.pdf">Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20041010092429/http://publish.uwo.ca/~harshman/wpppfac0.pdf">Archived</a> 2004-10-10 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. UCLA Working Papers in Phonetics, 16, pp. 1–84, 1970.

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<li id="cite_note-23"><a href="#cite_ref-23">^</a> L. D. Lathauwer, B. D. Moor, J. Vandewalle, <a rel="nofollow" class="external text" href="http://portal.acm.org/citation.cfm?id=354405">On the best rank-1 and rank-(R1, R2, ..., RN ) approximation of higher-order tensors</a>, SIAM Journal of Matrix Analysis and Applications 21 (4) (2000) 1324–1342.

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