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.[1]

In multilinear algebra, mode-m flattening[1][2][3], also known as matrixizing, matricizing, or unfolding,[4] is an operation that reshapes a multi-way array into a matrix denoted by (a two-way array).

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

Definition

The mode-m matrixizing of tensor is defined as the matrix . 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]

where and

By comparison, the matrix 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.

Applications

This operation is used in tensor algebra and its methods, such as Parafac and HOSVD.

References

  1. 1 2 3 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- 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. 1 2 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
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