Constructing Fast Approximate Eigenspaces With Application to the Fast Graph Fourier Transforms

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.