Tensor Completion by Multi-Rank via Unitary Transformation
This work provides a new theoretical bound for the number of samples needed for tensor completion, which is important for researchers working with high-dimensional data recovery.
This paper addresses the problem of determining the number of uniformly random sample entries required for tensor completion. It proposes a new bound for $n_1 \times n_2 \times n_3$ third-order tensor completion, utilizing the multi-rank of the tensor instead of its tubal rank.
One of the key problems in tensor completion is the number of uniformly random sample entries required for recovery guarantee. The main aim of this paper is to study $n_1 \times n_2 \times n_3$ third-order tensor completion based on transformed tensor singular value decomposition, and provide a bound on the number of required sample entries. Our approach is to make use of the multi-rank of the underlying tensor instead of its tubal rank in the bound. In numerical experiments on synthetic and imaging data sets, we demonstrate the effectiveness of our proposed bound for the number of sample entries. Moreover, our theoretical results are valid to any unitary transformation applied to $n_3$-dimension under transformed tensor singular value decomposition.