Twin support vector machine (TWSVM) is an efficient algorithm for binary classification. However, the lack of the structural risk minimization principle restrains the generalization of TWSVM and the guarantee of convex optimization constraints TWSVM to only use positive semi-definite kernels (PSD). In this paper, we propose a novel TWSVM for indefinite kernel called indefinite twin support vector machine with difference of convex functions programming (ITWSVM-DC). The indefinite TWSVM (ITWSVM) leverages a maximum margin regularization term to improve the generalization of TWSVM and a smooth quadratic hinge loss function to make the model continuously differentiable. The representer theorem is applied to the ITWSVM and the convexity of the ITWSVM is analyzed. In order to address the non-convex optimization problem when the kernel is indefinite, a difference of convex functions (DC) is used to decompose the non-convex objective function into the subtraction of two convex functions and a line search method is applied in the DC algorithm to accelerate the convergence rate. A theoretical analysis illustrates that ITWSVM-DC can converge to a local optimum and extensive experiments on indefinite and positive semi-definite kernels show the superiority of ITWSVM-DC.

双支持向量机（TWSVM）是一种有效的二元分类算法。然而，结构风险最小化原则的缺乏限制了 TWSVM 的泛化和凸优化约束 TWSVM 仅使用正半定核 (PSD) 的保证。在本文中，我们提出了一种用于不定核的新型 TWSVM，称为具有凸函数差异规划的不定孪生支持向量机（ITWSVM-DC）。不定 TWSVM (ITWSVM) 利用最大边际正则化项来提高 TWSVM 的泛化能力，并利用平滑的二次铰链损失函数使模型连续可微。将表示定理应用于ITWSVM，分析了ITWSVM的凸性。为了解决核不定时的非凸优化问题，利用凸函数差(DC)将非凸目标函数分解为两个凸函数相减，DC算法采用线搜索方法加快收敛速度​​。理论分析表明，ITWSVM-DC 可以收敛到局部最优，在不定和半正定核上的大量实验显示了 ITWSVM-DC 的优越性。