NFFT meets Krylov methods: Fast matrix-vector products for the graph Laplacian of fully connected networks

TL;DR

This paper introduces a fast matrix-vector multiplication method for the graph Laplacian using NFFT, enabling efficient eigenvalue computations on large, dense graphs without forming the entire matrix, applicable to various data science tasks.

Contribution

It develops a novel NFFT-based approach for fast matrix-vector products with the graph Laplacian, improving scalability and efficiency over traditional methods.

Findings

NFFT-based method outperforms Nyström method in speed and accuracy

Efficient eigenvalue computation for large dense graphs

Applicable to image segmentation and semi-supervised learning

Abstract

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 Nystr\"om method. Moreover, we present and test an enhanced, hybrid version of the Nystr\"om method, which internally uses the NFFT.