Efficient Low Average Rank Tensor Recovery With Implicit Low-Rank Subspace Regularization

Recent tensor methods utilizing the tensor nuclear norm (TNN) have shown remarkable performance in recovering low average rank tensor components. Nevertheless, methods exclusively dependent on the low average rank model encounter two practical challenges: 1) They overlook the significant low-rank attributes inherent in individual tensor modes, which are crucial for interpreting real-world data like color images and videos; 2) The minimization of TNN incurs a high computational cost due to the need for multiple tensor SVDs (t-SVDs). In response to the first challenge, our study introduces three key properties that elucidate the relationship between the Tucker rank and the average rank of tensors to highlight the fundamental distinctions between these two ranks. To address the second challenge, we introduce a novel tensor decomposition framework that facilitates the achievement of a low average rank while simultaneously leveraging the low-rank subspace prior. Importantly, we demonstrate that for a tensor possessing both low Tucker and average ranks, minimizing the TNN of a smaller core tensor within our model is equivalent to minimizing the TNN of the original tensor, which would reduce computational costs significantly. This nice property demonstrates proposed tensor decomposition method could simultaneously address the two challenges subtly. Additionally, we develop a non-convex augmented Lagrangian alternating direction minimization algorithm to solve our model and establish its convergence. Experiments on both synthetic and real-world datasets validate the effectiveness of our approach and https://faculty.uestc.edu.cn/gaobin/zh_CN/lwcg/153392/list/index.htm demo code is available: https://faculty.uestc.edu.cn/gaobin/zh_CN/lwcg/153392/list/index.htm.