Singular value decomposition of third order quaternion tensors

Abstract

A novel tensor product for third order quaternion tensors is proposed. With this tensor product, we further define the singular value decomposition and the rank of a third order quaternion tensor. It is also proved that the best rank-k approximation of a third order quaternion tensor exists. On the other hand, note that a color video can be expressed as a third order quaternion tensor, the proposed singular value decomposition thus opens up a new way to analyze the structure of color video. Finally, some experiments are performed to demonstrate the low-rankness of color videos.