  Much like the classical Fisher linear discriminant analysis (LDA), the
recently proposed Wasserstein discriminant analysis (WDA) is a linear
dimensionality reduction method that seeks a projection matrix to maximize the
dispersion of different data classes and minimize the dispersion of same data
classes via a bi-level optimization. In contrast to LDA, WDA can account for
both global and local interconnections between data classes by using the
underlying principles of optimal transport. In this paper, a bi-level nonlinear
eigenvector algorithm (WDA-nepv) is presented to fully exploit the structures
of the bi-level optimization of WDA. The inner level of WDA-nepv for computing
the optimal transport matrices is formulated as an eigenvector-dependent
nonlinear eigenvalue problem (NEPv), and meanwhile, the outer level for trace
ratio optimizations is formulated as another NEPv. Both NEPvs can be computed
efficiently under the self-consistent field (SCF) framework. WDA-nepv is
derivative-free and surrogate-model-free when compared with existing
algorithms. Convergence analysis of the proposed WDA-nepv justifies the
utilization of the SCF for solving the bi-level optimization of WDA. Numerical
experiments with synthetic and real-life datasets demonstrate the
classification accuracy and scalability of WDA-nepv.
