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.

中文翻译：

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三阶四元数张量的奇异值分解

提出了一种新的三阶四元数张量的张量积。有了这个张量积，我们进一步定义了奇异值分解和三阶四元数张量的秩。也证明了最好的排名——$克$存在三阶四元数张量的近似值。另一方面，请注意，彩色视频可以表示为三阶四元数张量，所提出的奇异值分解因此开辟了一种分析彩色视频结构的新方法。最后，进行了一些实验来证明彩色视频的低秩。