论文标题
支持向量机和ra的定理
Support vector machines and Radon's theorem
论文作者
论文摘要
支持向量计算机（SVM）是一种算法，它找到了一个超平面，该超平面最佳地将标记的数据点分为$ \ mathbb {r}^n $，为正和负类。该分离超平面边缘的数据点称为支持向量。我们将支持向量的可能配置连接到radon的定理，该定理可以保证一组点可以将一组分为两个类（正和负）相交的两个类（正和负）。如果将正和负支撑矢量的凸壳投射到分离的超平面上，则当且仅当超平面最佳时，投影相交。此外，在特定类型的一般位置下，我们表明（a）支撑向量的投影凸壳恰好在一个点上相交，（b）支持向量在扰动下是稳定的，（c）最多有$ n+1 $ support vectors和（d）支持向量从2到$ n+n+1 $。最后，我们对随机生成的数据进行了研究支持向量及其配置的预期数量的计算机模拟。我们观察到，随着这种类型的随机生成数据的点之间的距离增加，具有较少支持向量的配置变得更有可能。
A support vector machine (SVM) is an algorithm that finds a hyperplane which optimally separates labeled data points in $\mathbb{R}^n$ into positive and negative classes. The data points on the margin of this separating hyperplane are called support vectors. We connect the possible configurations of support vectors to Radon's theorem, which provides guarantees for when a set of points can be divided into two classes (positive and negative) whose convex hulls intersect. If the convex hulls of the positive and negative support vectors are projected onto a separating hyperplane, then the projections intersect if and only if the hyperplane is optimal. Further, with a particular type of general position, we show that (a) the projected convex hulls of the support vectors intersect in exactly one point, (b) the support vectors are stable under perturbation, (c) there are at most $n+1$ support vectors, and (d) every number of support vectors from 2 up to $n+1$ is possible. Finally, we perform computer simulations studying the expected number of support vectors, and their configurations, for randomly generated data. We observe that as the distance between classes of points increases for this type of randomly generated data, configurations with fewer support vectors become more likely.
