Title: Theory and Algorithms for Nonlinear Eigenvector Problems with Affine-Linear Structures
Abstract: Eigenvector-dependent Nonlinear Eigenvalue Problems (NEPv) have long played critical roles in computational physics and chemistry and are becoming increasingly important in numerous data science applications. In this talk, we consider a class of NEPv where the coefficient matrices have a special affine-linear structure. One important origin of affine-linear NEPv is the Rayleigh-quotient-related optimization, including the trace-ratio optimization for dimension reduction and robust Rayleigh-quotient optimization for handling data uncertainties. We will establish variational characterizations for particular affine-linear NEPv, and then provide a geometric interpretation of a Self-Consistent Fields (SCF) iteration for solving the NEPv. The geometric interpretation reveals the global convergence of SCF in many cases and explains its potential non-convergence issues in others. New improvements to SCF, including the local acceleration schemes and the global verification techniques, are also discussed. Numerical experiments demonstrate the effectiveness of our approach.