Rayleigh quotient minimization for absolutely one-homogeneous functionals

Approximation of the least Rayleigh quotient for degree p homogeneous functionals

Article history: Received 13 September 2016 Accepted 28 February 2017 Available online 6 March 2017 Communicated by E. Milman MSC: 35A15 35B40 35K55 47J10

Robust Rayleigh quotient minimization and nonlinear eigenvalue problems

In this paper, we study the robust Rayleigh quotient optimization problems that arise when optimizing in worst-case the Rayleigh quotient of data matrices subject to uncertainties. We propose to solve such problem by exploiting its characterization as a nonlinear eigenvalue problem with eigenvector nonlinearity. With this approach, we can show that a commonly used iterative method can be diverg...

Spectral Decompositions Using One-Homogeneous Functionals

This paper discusses the use of absolutely one-homogeneous regularization functionals in a variational, scale space, and inverse scale space setting to define a nonlinear spectral decomposition of input data. We present several theoretical results that explain the relation between the different definitions. Additionally, results on the orthogonality of the decomposition, a Parsevaltype identity...

The Rayleigh Quotient Iteration

The Rayleigh Quotient Iteration (RQI) was developed for real symmetric matrices. Its rapid local convergence is due to the stationarity of the Rayleigh Quotient at an eigenvector. Its excellent global properties are due to the monotonie decrease in the norms of the residuals. These facts are established for normal matrices. Both properties fail for nonnormal matrices and no generalization of th...

A Comparison of Parallel Preconditioners for the Sparse Generalized Eigenvalue Problems by Rayleigh-Quotient Minimization