SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 756-783, March 2024.
Abstract. In this paper, we consider a low-rank tensor recovery problem. Based on the tensor singular value decomposition (t-SVD), we propose the ratio of the tensor nuclear norm and the tensor Frobenius norm (TNF) as a novel nonconvex surrogate of tensor’s tubal rank. The rationale of the proposed model for enforcing a low-rank structure is analyzed as its theoretical properties. Specifically, we introduce a null space property (NSP) type condition, under which a low-rank tensor is a local minimum for the proposed TNF recovery model. Numerically, we consider a low-rank tensor completion problem as a specific application of tensor recovery and employ the alternating direction method of multipliers (ADMM) to secure a model solution with guaranteed subsequential convergence under mild conditions. Extensive experiments demonstrate the superiority of our proposed model over state-of-the-art methods.

中文翻译：

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低阶张量恢复中的尺度不变松弛及其在张量补全中的应用

SIAM 影像科学杂志，第 17 卷，第 1 期，第 756-783 页，2024 年 3 月。
摘要。在本文中，我们考虑低秩张量恢复问题。基于张量奇异值分解（t-SVD），我们提出张量核范数与张量弗罗贝尼乌斯范数（TNF）的比率作为张量输卵管秩的新型非凸替代物。所提出的强制低阶结构模型的基本原理作为其理论特性进行了分析。具体来说，我们引入了零空间属性（NSP）类型条件，在该条件下，低秩张量是所提出的 TNF 恢复模型的局部最小值。在数值上，我们将低秩张量完成问题视为张量恢复的具体应用，并采用乘子交替方向法（ADMM）来确保在温和条件下保证后续收敛的模型解。大量的实验证明了我们提出的模型相对于最先进的方法的优越性。