Various image datasets appear naturally in the form of multi-dimensional arrays (hypermatrices), called tensors. Image with incomplete entries, which often can be formulated as the low-rank tensor completion problem, is practically important in order to process large size 3D array images both efficiently and effectively. To make the low-rank approximation being tractable, most current approaches make use of the lower convex envelope of tensor multi-rank function for the approximation. Due to the gap between the rank function and its lower convex envelope, the adoption of tensor nuclear norm may lead to the approximation of the corresponding tensor tubal-rank being insufficient. In this paper, we introduce a new nonconvex regularization approach, which can better capture the low-rank characteristics than the convex approach. By transforming the original problem to the Fourier domain, we formulate an equivalent optimization problem with more transparent tensor rank characteristics, whose explicit solution can be obtained under our framework. A minimization algorithm, associated with the augmented Lagrangian multipliers and the nonconvex regularizer, is established and is shown to be feasible. The constructed sequence converges to the desirable Karush-Kuhn-Tucker point, which is mathematically validated in detail. Extensive experimental results demonstrate that our proposed approach outperforms the existing state-of-the-art convex approaches consistently.

中文翻译：

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通过张量分解和非凸正则化方法完成 3-D 阵列图像数据

各种图像数据集自然地以多维数组（超矩阵）的形式出现，称为张量。具有不完整条目的图像，通常可以表述为低秩张量补全问题，对于有效地处理大尺寸 3D 阵列图像实际上很重要。为了使低秩逼近易于处理，大多数当前方法利用张量多秩函数的下凸包络进行逼近。由于秩函数与其下凸包络之间的差距，采用张量核范数可能会导致对应张量tubal-rank的逼近不足。在本文中，我们介绍了一种新的非凸正则化方法，它可以比凸方法更好地捕捉低秩特征。通过将原始问题转换为傅里叶域，我们制定了具有更透明张量秩特征的等效优化问题，其显式解决方案可以在我们的框架下获得。建立了与增强拉格朗日乘子和非凸正则化器相关的最小化算法，并证明是可行的。构建的序列收敛到理想的 Karush-Kuhn-Tucker 点，该点在数学上得到了详细验证。广泛的实验结果表明，我们提出的方法始终优于现有的最先进的凸方法。建立了与增强拉格朗日乘子和非凸正则化器相关的最小化算法，并证明是可行的。构建的序列收敛到理想的 Karush-Kuhn-Tucker 点，该点在数学上得到了详细验证。广泛的实验结果表明，我们提出的方法始终优于现有的最先进的凸方法。建立了与增强拉格朗日乘子和非凸正则化器相关的最小化算法，并证明是可行的。构建的序列收敛到理想的 Karush-Kuhn-Tucker 点，该点在数学上得到了详细验证。广泛的实验结果表明，我们提出的方法始终优于现有的最先进的凸方法。