Tensor ring rank determination using odd-dimensional unfolding.

While tensor ring (TR) decomposition methods have been extensively studied, the determination of TR-ranks remains a challenging problem, with existing methods being typically sensitive to the determination of the starting rank (i.e., the first rank to be optimized). Moreover, current methods often fail to adaptively determine TR-ranks in the presence of noisy and incomplete data, and exhibit computational inefficiencies when handling high-dimensional data. To address these issues, we propose an odd-dimensional unfolding method for the effective determination of TR-ranks. This is achieved by leveraging the symmetry of the TR model and the bound rank relationship in TR decomposition. In addition, we employ the singular value thresholding algorithm to facilitate the adaptive determination of TR-ranks and use randomized sketching techniques to enhance the efficiency and scalability of the method. Extensive experimental results in rank identification, data denoising, and completion demonstrate the potential of our method for a broad range of applications.