Rayleigh quotient minimization for absolutely one-homogeneous functionals. (31st May 2019)

• Record Type: Journal Article
• Authors: Feld, Tal; Aujol, Jean-François; Gilboa, Guy; Papadakis, Nicolas
• Is Part Of: Inverse problems. Volume 35:Number 6(2019)
• Publication Date: 2019-05-31
• Subjects: Rayleigh quotient -- Cheeger cut -- absolutely one-homogeneous -- nonlinear eigenfunctions -- calibrable sets -- total variation; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ab0cb2
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: In this paper we examine the problem of minimizing generalized Rayleigh quotients of the form, 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, 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 L 1 sphere, which approximates the Cheeger-cut problem. Experimental results show the applicability of such algorithms for clustering and classification of images.