On the generalized vectorization and its inverse

TL;DR

This paper generalizes the vectorization operation from 2-D matrices to higher-dimensional arrays and formally defines its inverse, expanding the mathematical tools available for tensor operations.

Contribution

It introduces a novel generalization of vectorization for tensors and provides a formal definition of its inverse, which was previously not well-established.

Findings

Generalized vectorization for higher-dimensional arrays

Mathematically defined inverse operation

Potential applications in tensor analysis

Abstract

Although the vectorization operation is known and well-defined, it is only defined for 2-D matrices, and its inverse isn't as well-popularized. This work proposes to generalize the vectorization to higher dimensions, and define mathematically its inverse operation.