Quaternion tensor singular value decomposition using a flexible transform-based approach

TL;DR

This paper introduces a novel transform-based tensor singular value decomposition method for higher-order quaternion tensors, enabling effective low-rank approximations especially in color video processing.

Contribution

It proposes a new $ ext{QT}$-product and TQt-SVD for $L$th-order quaternion tensors, extending tensor decomposition techniques to quaternion data.

Findings

Effective in approximating color videos with third-order quaternion tensors

Provides the best TQt-rank-$s$ approximation using orthogonal quaternion transformations

Demonstrates the method's effectiveness through experimental validation

Abstract

A flexible transform-based tensor product named $\star_{{\rm{QT}}}$-product for $L$th-order ($L\geq 3$) quaternion tensors is proposed. Based on the $\star_{{\rm{QT}}}$-product, we define the corresponding singular value decomposition named TQt-SVD and the rank named TQt-rank of the $L$th-order ($L\geq 3$) quaternion tensor. Furthermore, with orthogonal quaternion transformations, the TQt-SVD can provide the best TQt-rank-$s$ approximation of any $L$th-order ($L\geq 3$) quaternion tensor. In the experiments, we have verified the effectiveness of the proposed TQt-SVD in the application of the best TQt-rank-$s$ approximation for color videos represented by third-order quaternion tensors.