Mode-k flattening

Short description: Mathematical operation

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 ${𝒜}$ 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 ${𝒜}\in {\mathbb{ℂ}}^{I_0\times I_1\times \cdots \times I_M},$ is defined as the matrix ${\mathbf{𝐀}}_{[m]}\in {\mathbb{ℂ}}^{I_m\times (I_0\dots I_{m-1}{I}_{m+1}\dots I_M)}$. 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

$$[{\mathbf{𝐀}}_{[m]}{]}_{jk}=a_{i_1\dots i_m\dots i_M},$$ where $j=i_m$ and $$k=1+\sum _{\genfrac{}{}{0ex}{}{n=0}{n\ne m}}^M(i_n-1)\prod _{\genfrac{}{}{0ex}{}{\ell =0}{\ell \ne m}}^{n-1}{I}_{\ell }.$$ By comparison, the matrix ${\mathbf{𝐀}}_{[m]}\in {\mathbb{ℂ}}^{I_m\times (I_{m+1}\dots I_M{I}_0{I}_1\dots I_{m-1})}$ 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.