论文标题
通过Tensor网络和Tucker包装器完成张量
Tensor Completion via Tensor Networks with a Tucker Wrapper
论文作者
论文摘要
近年来，由于其在图像/视频介绍，高光谱数据恢复等中的应用，低排名张量的完成（LRTC）受到了相当大的关注。提出了不同的张量等级（例如CP，Tucker，Tensor Train/Ring等）的不同概念，因此提出了各种基于优化的数字方法的lrtc。但是，尚未提出基于张量的网络方法。在本文中，我们建议使用塔克包装器通过张量网络解决LRTC。在这里，“塔克包装器”的意思是，张量网络的最外部因子矩阵都是正顺序的。我们将LRTC提出为解决非线性方程系统的问题，而不是一个受约束的优化问题。然后采用两级替代最小二乘方法来更新未知因素。该方法的计算以张量矩阵乘法为主，可以有效地执行。同样，在适当的假设下，表明该方法以线性速率收敛到精确溶液。数值模拟表明，所提出的算法与最新方法相媲美。
In recent years, low-rank tensor completion (LRTC) has received considerable attention due to its applications in image/video inpainting, hyperspectral data recovery, etc. With different notions of tensor rank (e.g., CP, Tucker, tensor train/ring, etc.), various optimization based numerical methods are proposed to LRTC. However, tensor network based methods have not been proposed yet. In this paper, we propose to solve LRTC via tensor networks with a Tucker wrapper. Here by "Tucker wrapper" we mean that the outermost factor matrices of the tensor network are all orthonormal. We formulate LRTC as a problem of solving a system of nonlinear equations, rather than a constrained optimization problem. A two-level alternative least square method is then employed to update the unknown factors. The computation of the method is dominated by tensor matrix multiplications and can be efficiently performed. Also, under proper assumptions, it is shown that with high probability, the method converges to the exact solution at a linear rate. Numerical simulations show that the proposed algorithm is comparable with state-of-the-art methods.
