We show how, using linear-algebraic tools developed to prove Tverberg's theorem in combinatorial geometry, we can design new models of multi-class support vector machines (SVMs). These supervised learning protocols require fewer conditions to classify sets of points, and can be computed using existing binary SVM algorithms in higher-dimensional spaces, including soft-margin SVM algorithms. We describe how the theoretical guarantees of standard support vector machines transfer to these new classes of multi-class support vector machines. We give a new simple proof of a geometric characterization of support vectors for largest margin SVMs by Veelaert.

翻译：我们展示了如何利用为证明组合几何中Tverberg定理而开发的线性代数工具，设计多类支持向量机（SVM）的新模型。这些监督学习协议对点集分类所需的约束条件更少，并且可通过在高维空间中应用现有二分类SVM算法（包括软间隔SVM算法）进行计算。我们阐述了标准支持向量机的理论保证如何迁移至这些新型多类支持向量机。文中还给出了Veelaert关于最大间隔SVM支持向量几何刻画的一个简洁新证明。