Robust Rayleigh quotient minimization and nonlinear eigenvalue problems

We study the robust Rayleigh quotient optimization problem where the data matrices of the Rayleigh quotient are subject to uncertainties. We propose to solve such a problem by exploiting its characterization as a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv). For solving the NEPv, we show that a commonly used iterative method can be divergent due to a wrong ordering of the eigenvalues. Two strategies are introduced to address this issue: a spectral transformation based on nonlinear shifting and a reformulation using second-order derivatives. Numerical experiments for applications in robust generalized eigenvalue classification, robust common spatial pattern analysis, and robust linear discriminant analysis demonstrate the effectiveness of the proposed approaches.