We investigate numerically efficient approximations of eigenspaces associated with symmetric and general matrices. The eigenspaces are factored into a fixed number of fundamental components that can be efficiently manipulated which we consider to be extended orthogonal Givens or scaling and shear transformations. The number of these components controls the trade-off between approximation accuracy and the computational complexity of projecting on the eigenspaces. We write minimization problems for the single fundamental components and provide closed-form solutions. Then we propose algorithms that iteratively update all these components until convergence. We show results on random matrices and an application on the approximation of graph Fourier transforms for directed and undirected graphs.

中文翻译：

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应用快速图傅立叶变换构造快速近似特征空间

我们研究了与对称矩阵和一般矩阵相关的特征空间的数值有效近似。特征空间被分解为固定数量的可以有效操作的基本分量，我们认为它们是扩展的正交 Givens 或缩放和剪切变换。这些组件的数量控制了近似精度和投影到特征空间的计算复杂性之间的权衡。我们为单个基本组件编写最小化问题并提供封闭形式的解决方案。然后我们提出了迭代更新所有这些组件直到收敛的算法。我们展示了随机矩阵的结果以及对有向图和无向图的图傅立叶变换近似的应用。