Constructing fast approximate eigenspaces with application to the fast graph Fourier transforms
This work addresses computational efficiency in spectral methods for graph processing, but it appears incremental as it builds on existing factorization techniques.
The paper tackles the problem of approximating eigenspaces for symmetric and general matrices by factoring them into fundamental components like Givens transformations, balancing accuracy and computational complexity, and demonstrates results on random matrices and graph Fourier transforms.
We investigate numerically efficient approximations of eigenspaces associated to symmetric and general matrices. The eigenspaces are factored into a fixed number of fundamental components that can be efficiently manipulated (we consider 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 iterative 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.