论文标题
基于梯度的块坐标下降算法的收敛，用于非正交关节矩阵的对角线化
Convergence of gradient-based block coordinate descent algorithms for non-orthogonal joint approximate diagonalization of matrices
论文作者
论文摘要
在本文中，我们提出了一个基于梯度的块坐标下降（BCD-G）框架，以解决在复杂的Stiefel歧管和特殊线性基团的乘积上定义的矩阵的近似对角线化。我们选择基于Riemannian梯度的块优化而不是循环方式。为了更新复杂的Stiefel歧管中的第一个块变量，我们使用众所周知的线搜索下降方法。为了基于四种不同的基本变换来更新特殊线性组中的第二个块变量，我们构建了三个类：GLU，GQU和GU，然后获得三个BCD-G算法：BCD-GLU，BCD-GQU，BCD-GQU和BCD-GU。在迭代元素是有限的假设下，我们使用lojasiewicz梯度不等式建立了这三种算法的全局和弱收敛性。我们还提出了一个基于梯度的Jacobi型框架，以解决特殊线性组定义的矩阵的关节对角线。就像在BCD-G情况下一样，使用GLU和GQU类别的基本转换类别，我们专注于Jacobi-Glu和Jacobi-GQU算法，并建立其全球和弱收敛性。本文所述的所有算法和收敛结果也适用于实际情况。
In this paper, we propose a gradient-based block coordinate descent (BCD-G) framework to solve the joint approximate diagonalization of matrices defined on the product of the complex Stiefel manifold and the special linear group. Instead of the cyclic fashion, we choose a block optimization based on the Riemannian gradient. To update the first block variable in the complex Stiefel manifold, we use the well-known line search descent method. To update the second block variable in the special linear group, based on four kinds of different elementary transformations, we construct three classes: GLU, GQU and GU, and then get three BCD-G algorithms: BCD-GLU, BCD-GQU and BCD-GU. We establish the global and weak convergence of these three algorithms using the Łojasiewicz gradient inequality under the assumption that the iterates are bounded. We also propose a gradient-based Jacobi-type framework to solve the joint approximate diagonalization of matrices defined on the special linear group. As in the BCD-G case, using the GLU and GQU classes of elementary transformations, we focus on the Jacobi-GLU and Jacobi-GQU algorithms and establish their global and weak convergence. All the algorithms and convergence results described in this paper also apply to the real case.
