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<title>Tensor decomposition</title>

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Tensor decomposition

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In <a href="Multilinear_algebra" title="Multilinear algebra">multilinear algebra</a>, a tensor decomposition is any scheme for expressing a <a href="Tensor_(machine_learning)" title="Tensor (machine learning)">"data tensor"</a> (M-way array) as a sequence of elementary operations acting on other, often simpler tensors.<a href="#cite_note-VasilescuDSP-1">[1]</a><a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a> Many tensor decompositions generalize some <a href="Matrix_decomposition" title="Matrix decomposition">matrix decompositions</a>.<a href="#cite_note-4">[4]</a>

<a href="Tensors" class="mw-redirect" title="Tensors">Tensors</a> are generalizations of matrices to higher dimensions (or rather to higher orders, i.e. the higher number of dimensions) and can consequently be treated as multidimensional fields.<a href="#cite_note-VasilescuDSP-1">[1]</a><a href="#cite_note-5">[5]</a>

The main tensor decompositions are:

<ul><li><a href="Tensor_rank_decomposition" title="Tensor rank decomposition">Tensor rank decomposition</a>;<a href="#cite_note-6">[6]</a></li>

<li><a href="Higher-order_singular_value_decomposition" title="Higher-order singular value decomposition">Higher-order singular value decomposition</a>;<a href="#cite_note-7">[7]</a></li>

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

<li><a href="Matrix_product_state" title="Matrix product state">matrix product states</a>, and operators or tensor trains;</li>

<li>Online Tensor Decompositions<a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a><a href="#cite_note-ektagujral-10">[10]</a></li>

<li>hierarchical Tucker decomposition;<a href="#cite_note-Vasilescu2019-11">[11]</a></li>

<li>block term decomposition<a href="#cite_note-12">[12]</a><a href="#cite_note-13">[13]</a><a href="#cite_note-Vasilescu2019-11">[11]</a><a href="#cite_note-14">[14]</a></li></ul>

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

This section introduces basic notations and operations that are widely used in the field.

<caption>Table of symbols and their description.

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<th>Symbols</th>

<th>Definition

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {a,{\bf {a}},{\bf {a}}^{T},\mathbf {A} ,{\mathcal {A}}}}</annotation>

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</math><img src="./266afabd0c3ccc3d29d938787d59664e7b3ec22a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.277ex; height:3.009ex;" alt="{\displaystyle {a,{\bf {a}},{\bf {a}}^{T},\mathbf {A} ,{\mathcal {A}}}}" loading="lazy"></td>

<td>scalar, vector, row, matrix, tensor

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\bf {a}}={vec(.)}}</annotation>

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</math><img src="./8dd1b882ea3441abf9a800cc64353f59dc12ba57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.459ex; height:2.843ex;" alt="{\displaystyle {\bf {a}}={vec(.)}}" loading="lazy"></td>

<td>vectorizing either a matrix or a tensor

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<annotation encoding="application/x-tex">{\displaystyle {\bf {A}}_{[m]}}</annotation>

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</math><img src="./a46c562850f437aee34eff77f06f4f5c61ffba4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.609ex; height:3.009ex;" alt="{\displaystyle {\bf {A}}_{[m]}}" loading="lazy"></td>

<td>matrixized tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation>

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</math><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \times _{m}}</annotation>

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</math><img src="./8e01d8cead6b83e96f0737bfa9b22652c5819369.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.483ex; height:2.009ex;" alt="{\displaystyle \times _{m}}" loading="lazy"></td>

<td>mode-m product

</td></tr></tbody></table>

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

A multi-way graph with K perspectives is a collection of K matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {X_{1},X_{2}.....X_{K}}}">

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {X_{1},X_{2}.....X_{K}}}</annotation>

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</math><img src="./d65ab0b5b27f00cadc9ccd87050fa3560208043c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.778ex; height:2.509ex;" alt="{\displaystyle {X_{1},X_{2}.....X_{K}}}" loading="lazy"> with dimensions I × J (where I, J are the number of nodes). This collection of matrices is naturally represented as a tensor X of size I × J × K. In order to avoid overloading the term “dimension”, we call an I × J × K tensor a three “mode” tensor, where “modes” are the numbers of indices used to index the tensor.

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

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