论文标题
基于张量分解的低等级矩阵完成的方法及其在张量完成的应用
Tensor factorization based method for low rank matrix completion and its application on tensor completion
论文作者
论文摘要
低等级矩阵和张量的完成问题是通过使用低级结构来恢复不完整的两个和高阶数据。矩阵和张量完成问题中的基本问题是如何提高效率。为此，我们首先建立了矩阵等级和张量管等级之间的关系，然后将矩阵的完成问题重新制定为张量的完成问题。对于重新制定的张量完成问题，我们采用了基于张量分解算法的两阶段策略。这样，可以通过较小尺寸的一些矩阵计算来解决大尺寸的矩阵完成问题。对于第三阶张量完成问题，为了完全利用低级结构，我们引入了双管等级，该等级结合了管等级和模式3展开矩阵的等级。对于模式3展开矩阵等级，我们遵循矩阵完成的想法。基于此，我们建立了一个新颖的模型，并修改了基于张量分解的算法，以完成三阶张量的完成。广泛的数值实验表明，所提出的方法在准确性和运行时间方面都优于最先进的方法。
Low rank matrix and tensor completion problems are to recover the incomplete two and higher order data by using their low rank structures. The essential problem in the matrix and tensor completion problems is how to improve the efficiency. To this end, we first establish the relationship between matrix rank and tensor tubal rank, and then reformulate matrix completion problem as a tensor completion problem. For the reformulated tensor completion problem, we adopt a two-stage strategy based on tensor factorization algorithm. In this way, a matrix completion problem of big size can be solved via some matrix computations of smaller sizes. For a third order tensor completion problem, to fully exploit the low rank structures, we introduce the double tubal rank which combines the tubal rank and the rank of the mode-3 unfolding matrix. For the mode-3 unfolding matrix rank, we follow the idea of matrix completion. Based on this, we establish a novel model and modify the tensor factorization based algorithm for third order tensor completion. Extensive numerical experiments demonstrate that the proposed methods outperform state-of-the-art methods in terms of both accuracy and running time.
