Kernel Bayesian tensor ring decomposition for multiway data recovery.

Tensor ring (TR) decomposition has emerged as the prevailing method for tensor completion. Earlier approaches have situated TR decomposition within a probabilistic framework, yielding satisfactory outcomes. However, these methods ignore side information or are inherently incapable of leveraging it. In response to this challenge, we propose a variational inference-based kernel Bayesian TR (VKBTR) method that integrates side information, low-rankness, and sparse learning. By incorporating kernel matrices into the TR factors, we can effectively leverage the intrinsic properties of the data (e.g., the smoothness in images and videos) to improve performance across different tasks. Additionally, by introducing a sparsity-inducing hierarchical prior on the latent factors, the proposed method enables automatic selection of the TR rank. Leveraging the variational inference algorithm enables us to achieve the update of posterior parameters effectively. Extensive experiments conducted on synthetic data, color images, face images, and color video data have shown that, with the assistance of side information, VKBTR significantly improves performance in completion tasks compared to other state-of-the-art methods.