The projection twin support vector machine (PTSVM) is a potential tool for classification problem. However the loss function of PTSVM is hinge loss function which is a unbounded loss and not robust enough to outliers. In this work, a robust PTSVM (termed RSHPTSVM) is proposed based on rescaled square hinge loss (RSH-loss) to handle classification problem. A close relationship between RSH-loss and correntropy is established theoretically. The RSH-loss can be viewed as a correntropy-induced loss by a reproducing piecewise kernel. As such a correntropy loss, it has vastly different properties from hinge loss such as boundedness, robustness and nonconvexity. Moreover, RSH-loss is with higher order statistical information from samples. However the nonconvexity of RSHPTSVM makes it difficult to optimize, so that an efficient iterative optimization algorithm based on semi-quadratic optimization theory is proposed to solve RSHPTSVM, which can quickly converge to the optimal solution. Furthermore, we extend our RSHPTSVM from binary classification to multi-classification and propose a robust projection multi-birth support vector machine model (termed RSHPMBSVM). The proposed method is implemented on various datasets including three artificial datasets, UCI datasets, and a practical application dataset. The experiment results under no noise and label noise circumstance confirm the feasibility and effectiveness of the proposed methods.

投影孪生支持向量机（PTSVM）是分类问题的潜在工具。然而，PTSVM 的损失函数是铰链损失函数，它是一种无界损失，对异常值不够稳健。在这项工作中，基于重新缩放的平方铰链损失（RSH-loss）提出了一种鲁棒的 PTSVM（称为 RSHPTSVM）来处理分类问题。理论上建立了 RSH 损失和相关熵之间的密切关系。RSH 损失可以被视为由复制分段内核引起的相关熵损失。作为这样的相关熵损失，它具有与铰链损失截然不同的特性，例如有界性、鲁棒性和非凸性。此外，RSH-loss 具有更高阶的样本统计信息。然而 RSHPTSVM 的非凸性使其难以优化，从而提出一种基于半二次优化理论的高效迭代优化算法求解RSHPTSVM，可以快速收敛到最优解。此外，我们将 RSHPTSVM 从二元分类扩展到多分类，并提出了一个鲁棒的投影多出生支持向量机模型（称为 RSHPMSVM）。所提出的方法在各种数据集上实现，包括三个人工数据集、UCI 数据集和一个实际应用数据集。无噪声和标签噪声情况下的实验结果证实了所提出方法的可行性和有效性。我们将 RSHPTSVM 从二元分类扩展到多分类，并提出了一个鲁棒的投影多生支持向量机模型（称为 RSHPMSVM）。所提出的方法在各种数据集上实现，包括三个人工数据集、UCI 数据集和一个实际应用数据集。无噪声和标签噪声情况下的实验结果证实了所提出方法的可行性和有效性。我们将 RSHPTSVM 从二元分类扩展到多分类，并提出了一个鲁棒的投影多出生支持向量机模型（称为 RSHPMSVM）。所提出的方法在各种数据集上实现，包括三个人工数据集、UCI 数据集和一个实际应用数据集。无噪声和标签噪声情况下的实验结果证实了所提出方法的可行性和有效性。