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Much like the classical Fisher linear discriminant analysis (LDA), therecently proposed Wasserstein discriminant analysis (WDA) is a lineardimensionality reduction method that seeks a projection matrix to maximize thedispersion of different data classes and minimize the dispersion of same dataclasses via a bi-level optimization. In contrast to LDA, WDA can account forboth global and local interconnections between data classes by using theunderlying principles of optimal transport. In this paper, a bi-level nonlineareigenvector algorithm (WDA-nepv) is presented to fully exploit the structuresof the bi-level optimization of WDA. The inner level of WDA-nepv for computingthe optimal transport matrices is formulated as an eigenvector-dependentnonlinear eigenvalue problem (NEPv), and meanwhile, the outer level for traceratio optimizations is formulated as another NEPv. Both NEPvs can be computedefficiently under the self-consistent field (SCF) framework. WDA-nepv isderivative-free and surrogate-model-free when compared with existingalgorithms. Convergence analysis of the proposed WDA-nepv justifies theutilization of the SCF for solving the bi-level optimization of WDA. Numericalexperiments with synthetic and real-life datasets demonstrate theclassification accuracy and scalability of WDA-nepv.

A Bi-level Nonlinear Eigenvector Algorithm for Wasserstein Discriminant Analysis