Statistical guaranteed noisy tensor recovery by fusing low-rankness on all orientations in frequency–original domains

Abstract

Low-rank tensor recovery faces challenges in accurately defining the low-rankness of a tensor. Most existing definitions typically focus on one domain alone — either the original or frequency domain. Additionally, certain definitions often exhibit limitations in their sensitivity to orientation variation. To overcome these challenges, we define a novel tensor rank, the Orientation Invariant Hybrid Rank (OIHR). This rank fuses rank information across all orientations in both frequency and original domains. Employing its convex approximation, the Orientation Invariant Hybrid Nuclear Norm (OIHNN), we propose a general tensor recovery model. We further explore the statistical performance of the estimator based on this model, establishing a deterministic upper bound on the estimation error under generic noise. Furthermore, non-asymptotic upper bounds under Gaussian noise are separately derived for two specific cases: tensor compressive sensing and tensor completion. Finally, we propose the algorithm to solve the model. Extensive experiments on both synthetic and real data are conducted to validate the statistical guarantees and verify the effectiveness of our algorithm.