UC Davis

The dissertation focuses on nonlinear eigenvector algorithms for generalized Rayleigh quotient optimizations and their applications.A well-known result in linear algebra and optimization is the connection between the Rayleigh quotient optimization and a linear eigenvalue problem. Consequently, linear eigensolvers can solve the Rayleigh quotient optimization. Another known result is the characterization of the trace ratio optimization by an eigenvector-dependent nonlinear eigenvalue problem (NEPv). The NEPv formulation of the trace ratio optimization can be solved using the self-consistent field (SCF) iteration. However, in the case of addressing the generalized variants of the Rayleigh quotient optimization, it remains an active area of research. We explore the nonlinear generalization of the Rayleigh quotient optimization in the context of the robust common spatial pattern, an algorithm used for signal processing in brain-computer interface system. Within this framework, the nonlinear Rayleigh quotient optimization is associated with a NEPv. We propose to solve this NEPv using the SCF iteration. The numerical advantages of this approach over existing methods are demonstrated using real-world datasets. Afterwards, we discuss Wasserstein discriminant analysis, a bi-level optimization for dimensionality reduction that is formulated as a nonlinear trace ratio optimization. We present an eigenvector algorithm for Wasserstein discriminant analysis that leverages the NEPv formulations of the inner and outer optimizations. We demonstrate convergence and scalability of our proposed eigenvector algorithm for Wasserstein discriminant analysis.