Tensor completion recovers missing components of multi-way data. The existing methods use either the Tucker rank or the CANDECOMP/PARAFAC (CP) rank in low-rank tensor optimization for data completion. In fact, these two kinds of tensor ranks represent different high-dimensional data structures. In this paper, we propose to exploit the two kinds of data structures simultaneously for image recovery through jointly minimizing the CP rank and Tucker rank in the low-rank tensor approximation. We use the alternating direction method of multipliers (ADMM) to reformulate the optimization model with two tensor ranks into its two sub-problems, and each has only one tensor rank optimization. For the two main sub-problems in the ADMM, we apply rank-one tensor updating and weighted sum of matrix nuclear norms minimization methods to solve them, respectively. The numerical experiments on some image and video completion applications demonstrate that the proposed method is superior to the state-of-the-art methods.

中文翻译：

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用于估计图像数据中缺失分量的低 CP 秩和 Tucker 秩张量完成

张量完成恢复多路数据的缺失组件。现有方法在低秩张量优化中使用 Tucker 秩或 CANDECOMP/PARAFAC (CP) 秩来完成数据。实际上，这两种张量秩代表了不同的高维数据结构。在本文中，我们建议通过联合最小化低秩张量近似中的 CP 秩和 Tucker 秩，同时利用两种数据结构进行图像恢复。我们使用乘法器的交替方向法（ADMM）将具有两个张量等级的优化模型重构为两个子问题，每个子问题只有一个张量等级优化。对于 ADMM 中的两个主要子问题，我们分别应用秩一张量更新和矩阵核范数加权和最小化方法来解决它们。