# A Bi-level Nonlinear Eigenvector Algorithm for Wasserstein Discriminant Analysis

arXiv: https://arxiv.org/abs/2211.11891 · HTML: https://mlsanity.com/paper/2211.11891/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 67 references that are in the map.
- First posted on arXiv 21 Nov 2022.

## Builds on (most cited first, 5 of 5)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Wasserstein Discriminant Analysis](https://mlsanity.com/paper/1608.08063/) (2016)
- [Screening Sinkhorn Algorithm for Regularized Optimal Transport](https://mlsanity.com/paper/1906.08540/) (2019)

## Abstract (arXiv)

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.