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 has 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.
翻译:这项工作旨在从数字上构建精确的通勤矩阵,接近于给定的几乎通勤矩阵,这相当于共同近似的对角化问题。 我们首先证明几乎接近通勤矩阵一般具有几乎对齐的近乎共同的向导元体。 基于这一关键观察, 我们提出一个快速和稳健的矢量- 向量- 联合对角化( VJD) 算法, 该算法通过按顺序查找这些近似常见的对流源体来构建正向相似性的变化。 在这样做的时候, 我们考虑在单位域上的次优化问题, 我们为此提出一种具有严格趋同分析的里曼尼半纽顿法。 我们还讨论拟议的VJD算法的数值稳定性。 提供独立组件分析中应用的数值示例, 以揭示与 Huaxin Lin 的对角体之间的关系, 并证明我们的方法与当时的雅各基型联合对角化算法相比是有利的。