The matrix completion problem can be found in many applications such as classification, image inpainting and collaborative filtering. In recent years, the emerging field of graph signal processing (GSP) has shed new light on this problem, deriving the graph signal matrix completion problem which incorporates the correlation of data elements. The nuclear-norm based methods possess satisfactory recovery performance, while they suffer from high computational cost and usually have slow convergence rate. In this paper, we propose two new iterative algorithms for solving the nuclear-norm regularization based graph signal matrix completion (NRGSMC) problem. By adopting approximate diagonalization approaches to estimate singular value decomposition (SVD), we obtain two generalized Newton algorithms, the generalized Newton with truncated Jacobi method (GNTJM) and the generalized Newton with parallel truncated Jacobi method (GNPTJM). The proposed methods are with low complexity and fast convergence by using the second-order information associated with the problem. Numerical results on three real-world data sets demonstrate that our schemes have evidently faster convergence rate than the gradient method with exact SVD, while maintain the similar completion performance.

矩阵完成问题可以在许多应用中找到，例如分类，图像修复和协作过滤。近年来，图形信号处理（GSP）的新兴领域对此问题有了新的认识，得出了包含数据元素相关性的图形信号矩阵完成问题。基于核标准的方法具有令人满意的恢复性能，同时它们具有计算成本高且收敛速度慢的缺点。在本文中，我们提出了两种新的迭代算法，用于解决基于核规范的正则化图信号矩阵完成（NRGSMC）问题。通过采用近似对角化方法来估计奇异值分解（SVD），我们获得了两种广义牛顿算法，截断Jacobi方法的广义牛顿（GNTJM）和平行截断Jacobi方法的广义牛顿（GNPTJM）。通过使用与问题相关的二阶信息，所提出的方法具有较低的复杂度和快速的收敛性。在三个实际数据集上的数值结果表明，与采用精确SVD的梯度方法相比，我们的方案具有明显更快的收敛速度，同时保持了相似的完井性能。