Mode-k flattening

In multi-linear algebra, mode-k flattening[1] (also matricisation, matricizing, or unfolding) is an operation on tensor (a multi-dimensional array) ${\mathcal {A}}$ denoted by $A_{(k)}$ turning it into a matrix (a two-dimensional array).

Matricization may be regarded as a generalization of the mathematical concept of vectorization.

Matricization may be applied in connection with determination of the factors in the PARAFAC model.

Applications

This operation is used widely in HoSVD.

Also, there is a mode-k flattening representation of SVD.

References

Eldén, L.; Savas, B. (2009-01-01). "A Newton–Grassmann Method for Computing the Best Multilinear Rank-$(r_1,$ $r_2,$ $r_3)$ 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.

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[1] Eldén, L.; Savas, B. (2009-01-01). "A Newton–Grassmann Method for Computing the Best Multilinear Rank-$(r_1,$ $r_2,$ $r_3)$ 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.