Low-Rank Tensor Ring Alternating Least Squares With Tikhonov Regularization for Image Completion

Abstract

Tensor decomposition has become an important technology for recovering missing information in high-dimensional image and video data. Alternatively, spatial regularization has been incorporated into tensor models such as Tucker, tensor train (TT), and tensor ring (TR). Existing alternating least squares (ALS) algorithms are efficient but generally lack regularization. Although spatial regularization techniques have been successfully integrated into ALS-based TT, extending this strategy to TR remains unsolved because of the circular structure and the additional trace operation. This paper proposes a tensor ring alternating least squares with Tikhonov regularization subproblem (TRATR), a framework that incorporates Tikhonov regularization into the core-wise update of the TR-ALS. Extensive experiments on images and videos under different missing rates demonstrate consistent improvements in relative standard error (RSE), peak signal-to-noise ratio (PSNR), and structural similarity index (SSIM), while reducing the required rank to 63.6% of the rank and yielding up to 83.6% reduction in update complexity. These results highlight the potential of TRATR for large-scale and resource-constrained applications.