  Recently, a quaternion tensor product named Qt-product was proposed, and then
the singular value decomposition and the rank of a third-order quaternion
tensor were given. From a more applicable perspective, we extend the Qt-product
and propose a novel multiplication principle for third-order quaternion tensor
named gQt-product. With the gQt-product, we introduce a brand-new singular
value decomposition for third-order quaternion tensors named gQt-SVD and then
define gQt-rank and multi-gQt-rank. We prove that the optimal low-rank
approximation of a third-order quaternion tensor exists and some numerical
experiments demonstrate the low-rankness of color videos. So, we apply the
low-rank quaternion tensor completion to color video inpainting problems and
present alternating least-square algorithms to solve the proposed low gQt-rank
and multi-gQt-rank quaternion tensor completion models. The convergence
analyses of the proposed algorithms are established and some numerical
experiments on various color video datasets show the high recovery accuracy and
computational efficiency of our methods.
