基于秩一矩阵补全的张量补全算法研究

Tensor Completion Algorithm Based on Rank-one Matrix Completion

张量作为矩阵的一般化形式,具有强大的数据表达能力和应用场景。然而,在许多科学研究和实际应用中,由于采样的缺失或方法的限制,想要获得完整的张量是比较困难的。因此,张量补全是十分重要的研究问题。首先提出矩阵补全算法,该算法基于秩一矩阵的贪心搜索,并在Frank-Wolfe算法的基础上进行改进,将原问题转化为凸优化问题,然后将这个算法扩展至张量补全中,对张量的每一维度展开进行补全。数值实验表明该算法和已有算法相比有较好的结果。

As the generalized form of matrix, tensor is extremely strong in representing complex data and has wide applications. However, it is quite hard to acquire complete tensors due to the sampling missing or restrictions on methods in many research and practical situations. Therefore, tensor completion becomes significant research problem. This paper first proposes a matrix comple- tion algorithm that based on greedy search in the space of rank-one matrices. It also has improvements on conditional gradient meth- od and transforms the original problem into a convex one. After that, the matrix completion algorithm is applied to the expansion of the tensor on each dimension. Numerical experiments show that our algorithm has better results compared with existing ones.