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Front. Appl. Math. Stat. | doi: 10.3389/fams.2018.00061

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

  • 1Technische Universität Chemnitz, Germany

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 matrix6 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.

Keywords: Graph Laplacian, Lanczos method, Eigenvalues, Nonequispaced fast Fourier transform (NFFT), machine learning

Received: 07 Sep 2018; Accepted: 28 Nov 2018.

Edited by:

Michael Ng, Hong Kong Baptist University, Hong Kong

Reviewed by:

Hyenkyun Woo, Korea University of Technology and Education, South Korea
Chao Chen, Stanford University, United States
Sergiy Pereverzyev Jr., Innsbruck Medical University, Austria  

Copyright: © 2018 Alfke, Potts, Stoll and Volkmer. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Prof. Daniel Potts, Technische Universität Chemnitz, Chemnitz, 09111, Lower Saxony, Germany,