Mode-k flattening

The mode-m matrixizing of tensor ${\displaystyle {\mathcal {A}}\in {\mathbb {C} }^{I_{0}\times I_{1}\times \cdots \times I_{M}},}$ is defined as the matrix ${\displaystyle {\bf {A}}_{[m]}\in {\mathbb {C} }^{I_{m}\times (I_{0}\dots I_{m-1}I_{m+1}\dots I_{M})}}$. 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[1]

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

where ${\displaystyle j=i_{m}}$ and

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

By comparison, the matrix ${\displaystyle {\bf {A}}_{[m]}\in {\mathbb {C} }^{I_{m}\times (I_{m+1}\dots I_{M}I_{0}I_{1}\dots I_{m-1})}}$ that results from an unfolding[4] 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.

This operation is used in tensor algebra and its methods, such as Parafac and HOSVD.