Advertisement

Computing Nonlinear Eigenfunctions via Gradient Flow Extinction

  • Leon BungertEmail author
  • Martin Burger
  • Daniel Tenbrinck
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11603)

Abstract

In this work we investigate the computation of nonlinear eigenfunctions via the extinction profiles of gradient flows. We analyze a scheme that recursively subtracts such eigenfunctions from given data and show that this procedure yields a decomposition of the data into eigenfunctions in some cases as the 1-dimensional total variation, for instance. We discuss results of numerical experiments in which we use extinction profiles and the gradient flow for the task of spectral graph clustering as used, e.g., in machine learning applications.

Keywords

Nonlinear eigenfunctions Spectral decompositions Gradient flows Extinction profiles Graph clustering 

Notes

Acknowledgments

This work was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 777826 (NoMADS). LB and MB acknowledge further support by ERC via Grant EU FP7 – ERC Consolidator Grant 615216 LifeInverse.

References

  1. 1.
    Andreu, F., Caselles, V., Diaz, J.I., Mazón, J.M.: Some qualitative properties for the total variation flow. J. Funct. Anal. 188(2), 516–547 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aujol, J.-F., Gilboa, G., Papadakis, N.: Theoretical analysis of flows estimating eigenfunctions of one-homogeneous functionals for segmentation and clustering. SIAM J. Imaging Sci. 11, 1416–1440 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bühler, T., Hein, M.: Spectral clustering based on the graph \(p\)-laplacian. In: International Conference on Machine Learning, pp. 81–88 (2009)Google Scholar
  4. 4.
    Bungert, L., Burger, M.: Solution paths of variational regularization methods for inverse problems. Inverse Prob. (2019). IOP PublishingGoogle Scholar
  5. 5.
    Bungert, L., Burger, M., Chambolle, A., Novaga,M.: Nonlinear spectral decompositions by gradient flows of one-homogeneous functionals. arXiv preprint arXiv:1901.06979 (2019)
  6. 6.
    Burger, M., Eckardt, L., Gilboa, G., Moeller, M.: Spectral representations of one-homogeneous functionals. In: Aujol, J.-F., Nikolova, M., Papadakis, N. (eds.) SSVM 2015. LNCS, vol. 9087, pp. 16–27. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-18461-6_2CrossRefGoogle Scholar
  7. 7.
    Burger, M., Gilboa, G., Moeller, M., Eckardt, L., Cremers, D.: Spectral decompositions using one-homogeneous functionals. SIAM J. Imaging Sci. 9(3), 1374–1408 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Elmoataz, A., Toutain, M., Tenbrinck, D.: On the \(p\)-laplacian and \(\infty \)-laplacian on graphs with applications in image and data processing. SIAM J. Imaging Sci. 8(4), 2412–2451 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gilboa, G.: A total variation spectral framework for scale and texture analysis. SIAM J. Imaging Sci. 7(4), 1937–1961 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gilboa, G.: Nonlinear Eigenproblems in Image Processing and Computer Vision. ACVPR. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-75847-3CrossRefzbMATHGoogle Scholar
  12. 12.
    Meng, Z., Merkurjev, E., Koniges, A., Bertozzi, A.L.: Hyperspectral image classification using graph clustering methods. Image Process. On Line 7, 218–245 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ng, A.Y., Jordan, M.I., Weiss, Y.: On spectral clustering: analysis and an algorithm. In: Advances in neural information processing systems, pp. 849–856 (2002)Google Scholar
  14. 14.
    Nossek, R.Z., Gilboa, G.: Flows generating nonlinear eigenfunctions. J. Sci. Comput. 75(2), 859–888 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Schmidt, M.F., Benning, M., Schönlieb, C.-B.: Inverse scale space decomposition. Inverse Prob. 34(4), 045008 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22, 107 (2000). Departmental Papers (CIS)CrossRefGoogle Scholar
  17. 17.
    von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department MathematikUniversität Erlangen-NürnbergErlangenGermany

Personalised recommendations