Mode-k flattening

Flattening a (3rd-order) tensor. The tensor can be flattened in three ways to obtain matrices comprising its mode-0, mode-1, and mode-2 vectors.

In multilinear algebra, mode-m flattening ^{[1] [2] [3]} , also known as matrixizing, matricizing is an operation that reshapes a multi-way array ${\displaystyle {\mathcal {A}}}$ into a matrix denoted by ${\displaystyle A_{[m]}}$ (a two-way array).

Matrixizing may be regarded as a generalization of the mathematical concept of vectorizing.

Definition

The mode-m matrixizing of tensor ${\displaystyle {\mathcal {A}}\in {\mathbb {C} }^{I_{0}\times I_{1}\times \cdots \times I_{M}},}$ is defined as the matrix ${\displaystyle {\bf {A}}_{[m]}\in {\mathbb {C} }^{I_{m}\times (I_{0}\dots I_{m-1}I_{m+1}\dots I_{M})}}$. As the parenthetical ordering indicates, the mode-m column vectors are arranged by sweeping all the other mode indices through their ranges, with smaller mode indexes varying more rapidly than larger ones; thus ^[1]

$${\displaystyle [{\bf {A}}_{[m]}]_{jk}=a_{i_{1}\dots i_{m}\dots i_{M}},}$$ where ${\displaystyle j=i_{m}}$ and $${\displaystyle k=1+\sum _{n=0 \atop n\neq m}^{M}(i_{n}-1)\prod _{\ell =0 \atop \ell \neq m}^{n-1}I_{\ell }.}$$ By comparison, the matrix ${\displaystyle {\bf {A}}_{[m]}\in {\mathbb {C} }^{I_{m}\times (I_{m+1}\dots I_{M}I_{0}I_{1}\dots I_{m-1})}}$ that results from an unfolding ^[4] has columns that are the result of sweeping through all the modes in a circular manner beginning with mode m + 1 as seen in the parenthetical ordering. This is an inefficient way to matrixize.^{[ citation needed ]}

Applications

This operation is used in tensor algebra and its methods, such as Parafac and HOSVD.^{[ citation needed ]}

References

1. Vasilescu, M. Alex O. (2009), "Multilinear (Tensor) Algebraic Framework for Computer Graphics, Computer Vision and Machine Learning" (PDF), University of Toronto, p. 21
2. Vasilescu, M. Alex O.; Terzopoulos, Demetri (2002), "Multilinear Analysis of Image Ensembles: TensorFaces" , Computer Vision — ECCV 2002, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 447–460, doi:10.1007/3-540-47969-4_30, ISBN   978-3-540-43745-1 , retrieved 2023-03-15
3. Eldén, L.; Savas, B. (2009-01-01), "A Newton–Grassmann Method for Computing the Best Multilinear Rank-${\displaystyle (r_{1},r_{2},r_{3})}$ Approximation of a Tensor", SIAM Journal on Matrix Analysis and Applications, 31 (2): 248–271, CiteSeerX   10.1.1.151.8143 , doi:10.1137/070688316, ISSN   0895-4798
4. De Lathauwer, Lieven; De Mood, B.; Vandewalle, J. (2000), "A multilinear singular value decomposition" , SIAM Journal on Matrix Analysis and Applications, 21 (4): 1253–1278, doi:10.1137/S0895479896305696