几乎交换矩阵的向量式联合对角化
Vector-wise Joint Diagonalization of Almost Commuting Matrices
AI总结：
提出向量式联合对角化算法，通过依次寻找近似公共特征向量构造正交变换，解决几乎交换矩阵的精确交换逼近问题，优于雅可比型算法。
摘要：
本文旨在数值构造与给定的几乎交换矩阵接近的精确交换矩阵，这等价于联合近似对角化问题。我们首先证明，几乎交换矩阵在一般位置下具有近似公共特征向量，且这些向量彼此几乎正交。基于这一关键观察，我们提出了一种快速且鲁棒的向量式联合对角化（VJD）算法，该算法通过依次寻找这些近似公共特征向量来构造正交相似变换。为此，我们考虑了单位球面上的子优化问题，并为此提出了一种具有严格收敛性分析的黎曼拟牛顿方法。我们还讨论了所提出的VJD算法的数值稳定性。提供了应用于独立成分分析的数值示例，以揭示与胡安林定理的联系，并证明我们的方法与最先进的雅可比型联合对角化算法相比具有优势。
英文摘要：
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.
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