Mode-k flattening explained

l{A} into a matrix denoted by 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

{ l A} \in { C} ^{I _{0 x} I _{1 x … x} I _{M}} , is defined as the matrix {\bf A} _{[m]} \in { C} ^{I _{m} x (I _{0} ... I _{m-1} I _{m+1} ... 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

$$[\{\backslash bf\; A\}\_\{[m]\}]\_\; =\; a\_,$$where

j = i _{m} and$$k=1+\backslash sum\_^M(i\_n\; -\; 1)\; \backslash prod\_^\; I\_\backslash ell.$$By comparison, the matrix {\bf A} _{[m]} \in { C} ^{I _{m} x (I _{m+1} ... I _{M} I _{0I} _{1} ... I _{m-1} )} 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 and HOSVD.