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.

翻译：与经典的Fisher线性判别分析（LDA）类似，近期提出的Wasserstein判别分析（WDA）是一种通过双层优化寻求投影矩阵的线性降维方法，其目标在于最大化不同数据类别的离散度并最小化同类数据类别的离散度。与LDA不同的是，WDA通过利用最优传输的基本原理可以同时考虑数据类别间的全局与局部相互关联。本文提出了一种双层非线性特征向量算法（WDA-nepv），以充分利用WDA双层优化的结构特性。WDA-nepv的内层计算最优传输矩阵被表述为特征向量依赖的非线性特征值问题（NEPv），同时外层用于迹比优化的部分被表述为另一个NEPv。这两个NEPv均可在自洽场（SCF）框架下高效求解。与现有算法相比，WDA-nepv无需导数且无需替代模型。对提出的WDA-nepv的收敛性分析论证了利用SCF求解WDA双层优化的合理性。基于合成数据集和真实数据集的数值实验验证了WDA-nepv的分类精度与可扩展性。