支持向量机与Radon定理
Support vector machines and Radon's theorem
摘要：
支持向量机（SVM）是一种算法，其找到将$\mathbb{R}^n$中的标记数据点最佳地分成正类和负类的超平面。在这个分离超平面的边缘上的数据点被称为支持向量。我们连接的可能配置的支持向量的Radon定理，它提供了保证时，一组点可以分为两类（正和负）的凸包相交。如果正、负支持向量的凸包投影到一个分离超平面上，则投影相交当且仅当该超平面是最优的。此外，与一个特定类型的一般位置，我们表明，（a）的投影凸包的支持向量相交在一个点，（b）的支持向量是稳定的扰动下，（c）有最多$n+1$支持向量，和（d）从2到$n+1$的支持向量的任何数量是可能的。最后，我们进行计算机模拟研究的预期数量的支持向量，和他们的配置，随机生成的数据。我们观察到，对于这种类型的随机生成的数据，随着点类之间的距离增加，具有较少支持向量的配置变得更有可能。
英文摘要：
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.
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