论文标题
几乎通勤矩阵的矢量关节对角线
Vector-wise Joint Diagonalization of Almost Commuting Matrices
论文作者
论文摘要
这项工作旨在在数值上构建几乎几乎通勤的矩阵，这等同于关节近似对角线问题。我们首先证明几乎通勤的矩阵具有近似近似彼此正交的近似常见特征向量。基于这一关键观察，我们提出了一种快速，健壮的矢量对角线化（VJD）算法，该算法通过顺序找到这些近似共同的特征向量来构建正交相似性变换。在此过程中，我们考虑了单位球体上的子优化问题，为此，我们提出了一种具有严格合并分析的Riemannian Quasi-Newton方法。我们还讨论了提出的VJD算法的数值稳定性。提供了在独立组件分析中应用的数值示例，以揭示与Huaxin Lin定理的关系，并证明我们的方法与最新的Jacobi-Type关节对角线化算法有利。
This work aims to numerically construct exactly commuting matrices close to given almost commuting ones, which is equivalent to the joint approximate diagonalization problem. We first prove that almost commuting matrices generically have approximate common eigenvectors that are almost orthogonal to each other. Based on this key observation, we propose a fast and robust vector-wise joint diagonalization (VJD) algorithm, which constructs the orthogonal similarity transform by sequentially finding these approximate common eigenvectors. In doing so, we consider sub-optimization problems over the unit sphere, for which we present a Riemannian quasi-Newton method with rigorous convergence analysis. We also discuss the numerical stability of the proposed VJD algorithm. Numerical examples with applications in independent component analysis are provided to reveal the relation with Huaxin Lin's theorem and to demonstrate that our method compares favorably with the state-of-the-art Jacobi-type joint diagonalization algorithm.
