This paper focuses on a specific family of classifiers called nonparallel
support vector classifiers (NPSVCs). Different from typical classifiers, the
training of an NPSVC involves the minimization of multiple objectives,
resulting in the potential concerns of feature suboptimality and class
dependency. Consequently, no effective learning scheme has been established to
improve NPSVCs' performance through representation learning, especially deep
learning. To break this bottleneck, we develop NPSVC++ based on multi-objective
optimization, enabling the end-to-end learning of NPSVC and its features. By
pursuing Pareto optimality, NPSVC++ theoretically ensures feature optimality
across classes, hence effectively overcoming the two issues above. A general
learning procedure via duality optimization is proposed, based on which we
provide two applicable instances, K-NPSVC++ and D-NPSVC++. The experiments show
their superiority over the existing methods and verify the efficacy of NPSVC++.

基于多目标优化，我们开发了 NPSVC++，实现了 NPSVC 及其特征的端到端学习，通过追求帕累托最优性，NPSVC++ 理论上确保了特征在各类别间的最优性，有效地克服了特征次优性和类别依赖性两个问题，实验表明其优于现有方法，验证了 NPSVC++ 的有效性。