Mode-k Flattening
In multilinear algebra, mode-m flattening, also known as matrixizing, matricizing, or unfolding, is an operation that reshapes a multi-way array \mathcal into a matrix denoted by A_ (a two-way array). Matrixizing may be regarded as a generalization of the mathematical concept of Vectorization (mathematics), vectorizing. Definition The mode-''m'' matrixizing of tensor \in ^, is defined as the matrix _ \in ^. 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 [_]_ = a_, where j=i_m and k=1+\sum_^M(i_n - 1) \prod_^ I_\ell. By comparison, the matrix _ \in ^ 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 ...
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