This study aims to solve the over-reliance on the rank estimation strategy in the standard tensor factorization-based tensor recovery and the problem of a large computational cost in the standard t-SVD-based tensor recovery. To this end, we proposes a new tensor norm with a dual low-rank constraint, which utilizes the low-rank prior and rank information at the same time. In the proposed tensor norm, a series of surrogate functions of the tensor tubal rank can be used to achieve better performance in harness low-rankness within tensor data. It is proven theoretically that the resulting tensor completion model can effectively avoid performance degradation caused by inaccurate rank estimation. Meanwhile, attributed to the proposed dual low-rank constraint, the t-SVD of a smaller tensor instead of the original big one is computed by using a sample trick. Based on this, the total cost at each iteration of the optimization algorithm is reduced to $\mathcal{O}(n^3\log n +kn^3)$ from $\mathcal{O}(n^4)$ achieved with standard methods, where $k$ is the estimation of the true tensor rank and far less than $n$. Our method was evaluated on synthetic and real-world data, and it demonstrated superior performance and efficiency over several existing state-of-the-art tensor completion methods.

翻译：本研究旨在解决标准张量分解类张量恢复方法中秩估计策略的过度依赖问题，以及标准t-SVD类张量恢复方法计算代价过大的问题。为此，我们提出了一种具有双重低秩约束的新型张量范数，该范数能同时利用低秩先验与秩信息。在该张量范数中，可采用一系列张量管状秩的代理函数，以更有效地利用张量数据中的低秩性实现更优性能。理论证明，由此导出的张量补全模型能有效避免因秩估计不准确导致的性能退化。同时，得益于所提出的双重低秩约束，通过采样技巧可计算较小张量（而非原始大张量）的t-SVD。据此，优化算法每次迭代的总计算代价从标准方法的$\mathcal{O}(n^4)$降低至$\mathcal{O}(n^3\log n +kn^3)$，其中$k$为真实张量秩的估计值且远小于$n$。在合成数据与真实数据上的实验表明，本文方法在性能与效率上均优于现有几种先进张量补全方法。