Toeplitz矩阵填充的四种流形逼近算法比较

 本文提出Toeplitz矩阵填充的四种流形逼近算法。在左奇异向量空间中对已知部分运用最小二乘法逼近，形成新的可行矩阵；并将对角线上的元素分别用均值，l₁范数，l_∞范数和中间数四种方法逼近使得迭代后的矩阵仍保持Toeplitz结构，节约了奇异向量空间的分解时间。最终找到合理的低秩矩阵来逼近未知的高秩矩阵，进而精确地完成Toeplitz矩阵的填充。理论上，分析了在一定条件下算法的收敛性。实验上，通过取不同的采样密度进行数值实验展示了四种算法的优劣。实验结果说明均值算法和l_∞范数算法大多用的时间较少，但是当采样密度和矩阵规模较大时，中间数算法的精度较高。

THE COMPARISON OF FOUR MANIFOLD APPROXIMATION ALGORITHMS FOR TOEPLITZ MATRIX COMPLETION

This paper proposes four manifold approximation algorithms for toeplitz matrix completion. A new feasible matrix is formed by using the least squares method in the left singular vector space. And the elements on the diagonal are approximated by the mean value, the l₁ norm, the l_∞ norm and the intermediate value to make the iterative matrix keep the Toeplitz structure. So that, the algorithm reduces the time of decomposition in the singular vector space. Finally find a reasonable low rank matrix to approximate the unknown high rank matrix, and then accurately complete the Toeplitz matrix completion. In theory, the convergence of the algorithm under certain conditions is analyzed. In experiment, numerical experiments are carried out by taking different sampling densities to show the advantages and disadvantages of the four algorithms. The experimental results show that the mean value algorithm and the l_∞ norm algorithm are mostly used for less time, but the precision of the intermediate value algorithm is higher when the sampling density and matrix size are large.