Robust Tensor SVD and Recovery With Rank Estimation

Tensor singular value decomposition (t-SVD) has recently become increasingly popular for tensor recovery under partial and/or corrupted observations. However, the existing
$t$
-SVD-based methods neither make use of a rank prior nor provide an accurate rank estimation (RE), which would limit their recovery performance. From the practical perspective, the tensor RE problem is nontrivial and difficult to solve. In this article, we, therefore, aim to
determine the correct rank
of an intrinsic low-rank tensor from corrupted observations based on t-SVD and further
improve recovery results with the estimated rank
. Specifically, we first induce the equivalence of the tensor nuclear norm (TNN) of a tensor and its
$f$
-diagonal tensor. We then simultaneously minimize the reconstruction error and TNN of the
$f$
-diagonal tensor, leading to RE. Subsequently, we relax our model by removing the TNN regularizer to improve the recovery performance. Furthermore, we consider more general cases in the presence of missing data and/or gross corruptions by proposing robust tensor principal component analysis and robust tensor completion with RE. The robust methods can achieve successful recovery by refining the models with correct estimated ranks. Experimental results show that the proposed methods outperform the state-of-the-art methods with significant improvements.