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, also known as matrixizing, matricizing, or unfolding, is an operation that reshapes a multi-way array ${\mathcal {A}}$ 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 ${\mathcal {A}}\in {\mathbb {C} }^{I_{0}\times I_{1}\times \cdots \times I_{M}},$ is defined as the matrix ${\bf {A}}_{[m]}\in {\mathbb {C} }^{I_{m}\times (I_{0}\dots I_{m-1}I_{m+1}\dots I_{M})}$. 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

$$[{\bf {A}}_{[m]}]_{jk}=a_{i_{1}\dots i_{m}\dots i_{M}},$$

where $j=i_{m}$ and

$$k=1+\sum _{n=0 \atop n\neq m}^{M}(i_{n}-1)\prod _{\ell =0 \atop \ell \neq m}^{n-1}I_{\ell }.$$

By comparison, the matrix ${\bf {A}}_{[m]}\in {\mathbb {C} }^{I_{m}\times (I_{m+1}\dots I_{M}I_{0}I_{1}\dots I_{m-1})}$ that results from an unfolding 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.