分层张量环完备化
Hierarchical Tensor Ring Completion
摘要：
张量完备化可以从高阶数据的部分观测项中估计出数据的缺失值。最近的工作表明，低阶张量环逼近是解决张量完备化问题的最有力的工具之一。然而，现有的算法需要预定义的张量环秩，这在实践中可能很难确定。为了解决这个问题，我们提出了一个层次张量环分解更紧凑的表示。我们在第一层使用标准张量环将张量分解为若干个3阶子张量，在第二层对每个子张量进一步进行张量奇异值分解（t-SVD）.在基于该分解的低秩张量完备化中，三阶核张量中的零元在第二层被修剪掉，这有助于自动确定张量环的秩.为了进一步提高恢复性能，我们使用全变差来利用局部分段平滑数据结构。乘子的交替方向法可以将最优化模型分解为若干个子问题，每一个子问题都可以高效求解。彩色图像和高光谱图像的数值实验表明，该算法优于国家的最先进的恢复精度。
英文摘要：
Tensor completion can estimate missing values of a high-order data from its partially observed entries. Recent works show that low rank tensor ring approximation is one of the most powerful tools to solve tensor completion problem. However, existing algorithms need predefined tensor ring rank which may be hard to determine in practice. To address the issue, we propose a hierarchical tensor ring decomposition for more compact representation. We use the standard tensor ring to decompose a tensor into several 3-order sub-tensors in the first layer, and each sub-tensor is further factorized by tensor singular value decomposition (t-SVD) in the second layer. In the low rank tensor completion based on the proposed decomposition, the zero elements in the 3-order core tensor are pruned in the second layer, which helps to automatically determinate the tensor ring rank. To further enhance the recovery performance, we use total variation to exploit the locally piece-wise smoothness data structure. The alternating direction method of multiplier can divide the optimization model into several subproblems, and each one can be solved efficiently. Numerical experiments on color images and hyperspectral images demonstrate that the proposed algorithm outperforms state-of-the-arts ones in terms of recovery accuracy.
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