Multidimensional Singular Value Decompositions Structure Comparison
Abstract
Singular value decomposition of matrices is an algorithm used to solve many problems, often considered as a single atomic operation. If a problem can be reduced to it and solved, then the problem can be considered as solved. In multidimensional cases that require singular value decomposition, generalizations such as tensor decompositions are employed. These decompositions are well-known and frequently used. However, they come with their own set of issues. Singular value decomposition of multidimensional matrices partially mitigates these problems. This article aims to visually illustrate the differences between the singular value decomposition of spatial matrices and the singular value decomposition of tensors. Structural and numerical distinctions will be presented, supported by clear examples. It will also highlight the drawbacks of existing tensor decompositions that are absent in the spatial matrix decomposition. In conclusion, tasks in which singular value decomposition of spatial matrices can replace singular value decomposition of tensors will be proposed.
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