In recent years, tensor networks have emerged as powerful tools for solving large-scale optimization problems. One of the most promising tensor networks is the tensor ring (TR) decomposition, which achieves circular dimensional permutation invariance in the model through the utilization of the trace operation and equitable treatment of the latent cores. On the other hand, more recently, quaternions have gained significant attention and have been widely utilized in color image processing tasks due to their effectiveness in encoding color pixels. Therefore, in this paper, we propose the quaternion tensor ring (QTR) decomposition, which inherits the powerful and generalized representation abilities of the TR decomposition while leveraging the advantages of quaternions for color pixel representation. In addition to providing the definition of QTR decomposition and an algorithm for learning the QTR format, this paper also proposes a low-rank quaternion tensor completion (LRQTC) model and its algorithm for color image inpainting based on the QTR decomposition. Finally, extensive experiments on color image inpainting demonstrate that the proposed QTLRC method is highly competitive.

翻译：近年来，张量网络已成为解决大规模优化问题的有力工具。其中最具有前景的张量网络之一是张量环（TR）分解，该分解通过利用迹运算并平等对待潜在核心，在模型中实现了循环维度置换不变性。另一方面，近年来四元数因其在编码彩色像素方面的有效性而受到广泛关注，并被广泛应用于彩色图像处理任务中。因此，本文提出四元数张量环（QTR）分解，该分解继承了TR分解强大且泛化的表示能力，同时利用了四元数在彩色像素表示方面的优势。除提供QTR分解的定义及学习QTR格式的算法外，本文还基于QTR分解提出了一种低秩四元数张量补全（LRQTC）模型及其用于彩色图像修复的算法。最后，在彩色图像修复任务上的大量实验表明，所提出的QTLRC方法具有高度竞争力。