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

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.Comment: 28 pages, 9 figure

Model order reduction for delay systems by iterative interpolation

AbstractAdaptive algorithms for computing the reduced‐order model of time‐delay systems (TDSs) are proposed in this work. The algorithms are based on interpolating the transfer function at multiple expansion points and greedy iterations for selecting the expansion points. The ‐error of the reduced transfer function is used as the criterion for choosing the next new expansion point. One heuristic greedy algorithm and one algorithm based on the error system and adaptive sub‐interval selection are developed. Results on four TDSs with tens of delays from electromagnetic applications are presented and show the efficiency of the proposed algorithms