Abstract
It has been recently shown that a large class of balanced graph cuts allows for an exact relaxation into a nonlinear eigenproblem. We review briefly some of these results and propose a family of algorithms to compute nonlinear eigenvectors which encompasses previous work as special cases. We provide a detailed analysis of the properties and the convergence behavior of these algorithms and then discuss their application in the area of balanced graph cuts.
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Notes
- 1.
A function \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}\) is (positively) p-homogeneous if A(ν x) = ν p A(x) for all \(\nu \in \mathbb{R}\) (ν ≥ 0). In the following we will call functions just homogeneous when referring to positive homogeneity.
References
Bach, F.: Learning with submodular functions: a convex optimization perspective. Found. Trends Mach. Learn. 6(2–3), 145–373 (2008)
Bresson, X., Laurent, T., Uminsky, D., von Brecht, J.H.: Convergence and energy landscape for cheeger cut clustering. In: Advances in Neural Information Processing Systems 25 (NIPS), pp. 1394–1402. Curran Associates, Red Hook (2012)
Bresson, X., Laurent, T., Uminsky, D., von Brecht, J.H.: Convergence of a steepest descent algorithm for ratio cut clustering (2012). ArXiv:1204.6545v1
Bühler, T., Hein, M.: Spectral clustering based on the graph p-Laplacian. In: Proceedings of the 26th International Conference on Machine Learning (ICML), Montreal, pp. 81–88 (2009)
Bühler, T., Rangapuram, S., Setzer, S., Hein, M.: Constrained fractional set programs and their application in local clustering and community detection. In: Proceedings of the 30th International Conference on Machine Learning (ICML), Atlanta, pp. 624–632 (2013)
Clarke, F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Guattery, S., Miller, G.L.: On the quality of spectral separators. SIAM J. Matrix Anal. Appl. 19, 701–719 (1998)
Hein, M., Bühler, T.: An inverse power method for nonlinear eigenproblems with applications in 1-spectral clustering and sparse PCA. In: Advances in Neural Information Processing Systems 23 (NIPS), pp. 847–855. Curran Associates, Red Hook (2010)
Hein, M., Setzer, S.: Beyond spectral clustering – tight relaxations of balanced graph cuts. In: Advances in Neural Information Processing Systems 24 (NIPS), pp. 2366–2374. Neural Information Processing Systems/Curran Associates, La Jolla/Red Hook (2011)
Hein, M., Setzer, S., Jost, L., Rangapuram, S.: The total variation on hypergraphs – learning on hypergraphs revisited. In: Advances in Neural Information Processing Systems 26 (NIPS), pp. 2427–2435 (2013)
Szlam, A., Bresson, X.: Total variation and Cheeger cuts. In: Proceedings of the 27th International Conference on Machine Learning (ICML), Haifa, pp. 1039–1046 (2010)
von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17, 395–416 (2007)
Walshaw, C.: The graph partitioning archive (2004). Staffweb.cms.gre.ac.uk/~c.walshaw/partition/
Yang, F., Wei, Z.: Generalized Euler identity for subdifferentials of homogeneous functions and applications. J. Math. Anal. Appl. 337, 516–523 (2008)
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Jost, L., Setzer, S., Hein, M. (2014). Nonlinear Eigenproblems in Data Analysis: Balanced Graph Cuts and the RatioDCA-Prox. In: Dahlke, S., et al. Extraction of Quantifiable Information from Complex Systems. Lecture Notes in Computational Science and Engineering, vol 102. Springer, Cham. https://doi.org/10.1007/978-3-319-08159-5_13
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DOI: https://doi.org/10.1007/978-3-319-08159-5_13
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