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Flows Generating Nonlinear Eigenfunctions

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Abstract

Linear eigenvalue analysis has provided a fundamental framework for many scientific and engineering disciplines. Consequently, vast research was devoted to numerical schemes for computing eigenfunctions. In recent years, new research in image processing and machine-learning has shown the applicability of nonlinear eigenvalue analysis, specifically based on operators induced by convex functionals. This has provided new insights, better theoretical understanding and improved image-processing, clustering and classification algorithms. However, the theory of nonlinear eigenvalue problems is still very preliminary. We present a new class of nonlinear flows that can generate nonlinear eigenfunctions of the form \(T(u)=\lambda u\), where T(u) is a nonlinear operator and \(\lambda \in \mathbb {R} \) is the eigenvalue. We develop the theory where T(u) is a subgradient element of a regularizing one-homogeneous functional, such as total-variation or total-generalized-variation. We focus on a forward flow which simultaneously smooths the solution (with respect to the regularizer) while increasing the 2-norm. An analog discrete flow and its normalized version are formulated and analyzed. The flows translate to a series of convex minimization steps. In addition we suggest an indicator to measure the affinity of a function to an eigenfunction and relate it to pseudo-eigenfunctions in the linear case.

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Acknowledgements

We would like to thank Martin Benning, Nicolas Papadakis and Jean-Francois Aujol for stimulating discussions and helpful comments. We would further like to thank the two anonymous reviewers for their suggestions. We acknowledge support by the Israel Science Foundation (Grant No. 718/15).

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  1. Department of Electrical Engineering, Technion - Israel Institute of Technology, 32000, Haifa, Israel

    Raz Z. Nossek & Guy Gilboa

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  1. Raz Z. Nossek
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Correspondence to Raz Z. Nossek.

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Nossek, R.Z., Gilboa, G. Flows Generating Nonlinear Eigenfunctions. J Sci Comput 75, 859–888 (2018). https://doi.org/10.1007/s10915-017-0577-6

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