Fast Eigenspace Approximation using Random Signals
This addresses the computational bottleneck in graph-based tasks like clustering and visualization for large-scale networks, offering a more scalable solution.
The paper tackles the problem of efficiently estimating the first k eigenvectors of a graph Laplacian by filtering Gaussian random signals, proving that only k signals are needed for exact recovery regardless of graph size, and demonstrates better scaling and almost perfect reconstruction in experiments.
We focus in this work on the estimation of the first $k$ eigenvectors of any graph Laplacian using filtering of Gaussian random signals. We prove that we only need $k$ such signals to be able to exactly recover as many of the smallest eigenvectors, regardless of the number of nodes in the graph. In addition, we address key issues in implementing the theoretical concepts in practice using accurate approximated methods. We also propose fast algorithms both for eigenspace approximation and for the determination of the $k$th smallest eigenvalue $λ_k$. The latter proves to be extremely efficient under the assumption of locally uniform distribution of the eigenvalue over the spectrum. Finally, we present experiments which show the validity of our method in practice and compare it to state-of-the-art methods for clustering and visualization both on synthetic small-scale datasets and larger real-world problems of millions of nodes. We show that our method allows a better scaling with the number of nodes than all previous methods while achieving an almost perfect reconstruction of the eigenspace formed by the first $k$ eigenvectors.