NFFT meets Krylov methods: Fast matrix-vector products for the graph Laplacian of fully connected networks
This addresses efficiency problems for researchers and practitioners in data science and machine learning dealing with large-scale graph-based computations, though it is incremental as it builds on existing NFFT and Krylov methods.
The paper tackles the computational challenge of performing matrix-vector products with densely populated graph Laplacians in large networks by using the nonequispaced fast Fourier transform (NFFT) to accelerate these operations without forming the full matrix, achieving feasibility in applications like image segmentation and semi-supervised learning, with comparisons to the Nyström method.
The graph Laplacian is a standard tool in data science, machine learning, and image processing. The corresponding matrix inherits the complex structure of the underlying network and is in certain applications densely populated. This makes computations, in particular matrix-vector products, with the graph Laplacian a hard task. A typical application is the computation of a number of its eigenvalues and eigenvectors. Standard methods become infeasible as the number of nodes in the graph is too large. We propose the use of the fast summation based on the nonequispaced fast Fourier transform (NFFT) to perform the dense matrix-vector product with the graph Laplacian fast without ever forming the whole matrix. The enormous flexibility of the NFFT algorithm allows us to embed the accelerated multiplication into Lanczos-based eigenvalues routines or iterative linear system solvers and even consider other than the standard Gaussian kernels. We illustrate the feasibility of our approach on a number of test problems from image segmentation to semi-supervised learning based on graph-based PDEs. In particular, we compare our approach with the Nyström method. Moreover, we present and test an enhanced, hybrid version of the Nyström method, which internally uses the NFFT.