In multilinear algebra, mode-m flattening, also known as matrixizing, matricizing, or unfolding, is an operation that reshapes a multi-way array

\mathcal

into a matrix denoted by

A_

(a two-way array). Matrixizing may be regarded as a generalization of the mathematical concept of vectorizing.

Definition

The mode-''m'' matrixizing of tensor

\in ^,

is defined as the matrix

_ \in ^

. 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

= a_,

where

j=i_m

and

k=1+\sum_^M(i_n - 1) \prod_^ I_\ell.

By comparison, the matrix

_ \in ^

that results from an ''unfolding'' has columns that are the result of sweeping through all the modes in a circular manner beginning with mode 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 In multilinear algebra, the tensor rank decomposition or the rank-R decomposition of a tensor is the decomposition of a tensor in terms of a sum of minimum R rank-1 tensors. This is an open problem. Canonical polyadic decomposition (CPD) is a var ...

and

HOSVD In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one generalization of the matrix singular value decomposition. It has applications in co ...

References

{{reflist, refs = {{citation , last = De Lathauwer , first = Lieven , last2 = De Mood , first2 = B. , last3 = Vandewalle , first3 = J. , date = 2000 , title = A multilinear singular value decomposition , journal = SIAM Journal on Matrix Analysis and Applications , volume = 21 , issue = 4 , pages = 1253-1278 , url = http://www.siam.org/journals/simax/21-4/30569.html {{citation , last = Eldén , first = L. , last2 = Savas , first2 = B. , date = 2009-01-01 , title = A Newton–Grassmann Method for Computing the Best Multilinear Rank-

(r_1, r_2, r_3)

Approximation of a Tensor , url = http://epubs.siam.org/doi/10.1137/070688316 , journal = SIAM Journal on Matrix Analysis and Applications , volume = 31 , issue = 2 , pages = 248–271 , doi = 10.1137/070688316 , issn = 0895-4798 , citeseerx = 10.1.1.151.8143 {{citation , last = Vasilescu , first = M. Alex O. , title = Multilinear (Tensor) Algebraic Framework for Computer Graphics, Computer Vision and Machine Learning , date = 2009 , url = https://tspace.library.utoronto.ca/bitstream/1807/65327/11/Vasilescu_M_Alex_O_200911_PhD_thesis.pdf , work = University of Toronto , page = 21 {{citation , last = Vasilescu , first = M. Alex O. , last2 = Terzopoulos , first2 = Demetri , title = Multilinear Analysis of Image Ensembles: TensorFaces , date = 2002 , url = http://dx.doi.org/10.1007/3-540-47969-4_30 , work = Computer Vision — ECCV 2002 , pages = 447–460 , access-date = 2023-03-15 , place = Berlin, Heidelberg , publisher = Springer Berlin Heidelberg , isbn = 978-3-540-43745-1 Algebra Multilinear algebra