ST-SVD Factorization and s-Diagonal Tensors

TL;DR

This paper introduces s-diagonal tensors obtained via a specific transformation involving DFT and SVD, characterizes their properties, and provides necessary and sufficient conditions for their identification in third order tensors.

Contribution

It defines s-diagonal tensors, characterizes their properties, and establishes necessary and sufficient conditions for their identification in third order tensor spaces.

Findings

Each orthogonal equivalence class has a unique s-diagonal tensor.

Third order tensors in the same class share tensor tubal rank and T-singular values.

Explicit conditions are provided for dimensions 2, 3, and 4.

Abstract

A third order real tensor is mapped to a special f-diagonal tensor by going through Discrete Fourier Transform (DFT), standard matrix SVD and inverse DFT. We call such an f-diagonal tensor an s-diagonal tensor. An f-diagonal tensor is an s-diagonal tensor if and only if it is mapped to itself in the above process. The third order tensor space is partitioned to orthogonal equivalence classes. Each orthogonal equivalence class has a unique s-diagonal tensor. Two s-diagonal tensors are equal if they are orthogonally equivalent. Third order tensors in an orthogonal equivalence class have the same tensor tubal rank and T-singular values. Four meaningful necessary conditions for s-diagonal tensors are presented. Then we present a set of sufficient and necessary conditions for s-diagonal tensors. Such conditions involve a special complex number. In the cases that the dimension of the third mode of the considered tensor is $2, 3$ and $4$, we present direct sufficient and necessary conditions which do not involve such a complex number.