  Recently, numerous tensor singular value decomposition (t-SVD)-based tensor
recovery methods have shown promise in processing visual data, such as color
images and videos. However, these methods often suffer from severe performance
degradation when confronted with tensor data exhibiting non-smooth changes. It
has been commonly observed in real-world scenarios but ignored by the
traditional t-SVD-based methods. In this work, we introduce a novel tensor
recovery model with a learnable tensor nuclear norm to address such a
challenge. We develop a new optimization algorithm named the Alternating
Proximal Multiplier Method (APMM) to iteratively solve the proposed tensor
completion model. Theoretical analysis demonstrates the convergence of the
proposed APMM to the Karush-Kuhn-Tucker (KKT) point of the optimization
problem. In addition, we propose a multi-objective tensor recovery framework
based on APMM to efficiently explore the correlations of tensor data across its
various dimensions, providing a new perspective on extending the t-SVD-based
method to higher-order tensor cases. Numerical experiments demonstrated the
effectiveness of the proposed method in tensor completion.
