论文标题
三阶张量完成的Riemannian共轭梯度下降法
Riemannian Conjugate Gradient Descent Method for Third-Order Tensor Completion
论文作者
论文摘要
张量完成的目的是填充在低级别约束下部分已知的张量的缺失条目。在本文中，我们主要通过在平滑歧管上使用Riemannian优化方法来研究低级三阶张量完成问题。在这里，张量秩被定义为一组矩阵等级，其中矩阵是通过将与傅立叶相关的转换应用于原始张量的试管中获得的转换张量的切片。我们表明，在基础低级张量的合适不相互性条件下，提出的Riemannian优化方法可以保证会收敛并发现具有很高概率的低级张量。此外，研究并得出了在不同初始化方法下解决低级张量完成问题所需的样本条目的数量。据报道，合成数据集和图像数据集的数值示例证明了所提出的方法能够恢复低级张量。
The goal of tensor completion is to fill in missing entries of a partially known tensor under a low-rank constraint. In this paper, we mainly study low rank third-order tensor completion problems by using Riemannian optimization methods on the smooth manifold. Here the tensor rank is defined to be a set of matrix ranks where the matrices are the slices of the transformed tensor obtained by applying the Fourier-related transformation onto the tubes of the original tensor. We show that with suitable incoherence conditions on the underlying low rank tensor, the proposed Riemannian optimization method is guaranteed to converge and find such low rank tensor with a high probability. In addition, numbers of sample entries required for solving low rank tensor completion problem under different initialized methods are studied and derived. Numerical examples for both synthetic and image data sets are reported to demonstrate the proposed method is able to recover low rank tensors.
