Abstract. Recently, quaternion-based methods have demonstrated their superiority over real-based methods in fields such as high-dimensional data compression and recovery. In this paper, we focus on low-rank quaternion tensor completion (LRQTC), which aims to recover the original high-dimensional data from incomplete observations. Within the framework of the tensor-tensor product of the quaternion form, this work achieves the following fourfold contributions: (i) We define several key concepts for quaternion tensors, including the quaternion tensor tubal rank and the quaternion tensor spectral norm, and further introduce the quaternion tensor tubal nuclear norm (QTNN). Theoretically, we prove that the QTNN serves as the convex envelope of the quaternion tensor average rank within the unit ball of the quaternion tensor spectral norm. (ii) We propose a novel LRQTC framework based on quaternion tensor tubal rank and further establish the exact recovery guarantee for LRQTC, demonstrating that under appropriate incoherence conditions, minimizing QTNN can exactly reconstruct a low-tubal-rank quaternion tensor with high probability. (iii) We develop an efficient quaternion tensor singular value thresholding algorithm, which provides a closed-form solution for the QTNN minimization problem. (iv) Extensive experiments on both synthetic and real-world data validate our theoretical analysis and demonstrate the superior performance of the proposed method over several state-of-the-art methods in terms of both quantitative metrics and visual quality.
Zheng et al. (Fri,) studied this question.