Rayleigh quotient minimization for absolutely one-homogeneous functionals

Abstract : In this paper we examine the problem of minimizing generalized Rayleigh quotients of the form J(u)/H(u), where both J and H are absolutely one-homogeneous functionals. This can be viewed as minimizing J where the solution is constrained to be on a generalized sphere with H(u) = 1, where H is any norm or semi-norm. The solution admits a nonlinear eigenvalue problem, based on the subgradients of J and H. We examine several flows which minimize the ratio. This is done both by time-continuous flow formulations and by discrete iterations. We focus on a certain flow, which is easier to analyze theoretically, following the theory of Brezis on flows with maximal monotone operators. A comprehensive theory is established, including convergence of the flow. We then turn into a more specific case of minimizing graph total variation on the L1 sphere, which approximates the Cheeger-cut problem. Experimental results show the applicability of such algorithms for clustering and classification of images.
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https://hal.archives-ouvertes.fr/hal-01864129
Contributor : Jean-François Aujol <>
Submitted on : Thursday, August 30, 2018 - 6:16:08 PM
Last modification on : Friday, March 8, 2019 - 1:14:20 AM

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Tal Feld, Jean-François Aujol, Guy Gilboa, Nicolas Papadakis. Rayleigh quotient minimization for absolutely one-homogeneous functionals. Inverse Problems, IOP Publishing, 2019, ⟨10.1088/1361-6420/ab0cb2⟩. ⟨hal-01864129v2⟩

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