NFFT Meets Krylov Methods
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 Nyström method. Moreover, we present and test an enhanced, hybrid version of the Nyström method, which internally uses the NFFT.
Publication: D. Alfke, D. Potts, M. Stoll, and T. Volkmer, NFFT Meets Krylov Methods: Fast Matrix-Vector Products for the Graph Laplacian of Fully Connected Networks. Frontiers in Applied Mathematics and Statistics, 4 (2018). Available online at https://doi.org/10.3389/fams.2018.00061.
Software: We provide the MATLAB software NFFT-Lanczos-Example showcasing the basic usage of our proposed method.
Chebyshev Semi-IterationIn our paper
Publication: All-at-once preconditioning in PDE-constrained optimization; Tyrone Rees; Martin Stoll; Andy Wathen; Kybernetika : Vol. 46 (2); 2010the algorithm listing the Chebyshev semi-iteration contains two errors. Line 6 and Line 8 should contain μ instead of ω with μ depending on the spectrum of the matrix.
Software: For an example code follow CSI Code .