核函数的性质及其构造方法

支持向量机是一项机器学习技术，发展至今近10年了，已经成功地用于模式识别、回归估计以及聚类等，并由此衍生出了核方法。支持向量机由核函数与训练集完全刻画。进一步提高支持向量机性能的关键，是针对给定的问题设计恰当的核函数，这就要求对核函数本身有深刻了解。本文首先分析了核函数的一些重要性质，接着对3类核函数，即平移不变核函数、旋转不变核函数和卷积核，提出了简单实用的判别准则。在此基础上，验证和构造了很多重要核函数。

Properties and Construction Methods of Kernel in Support Vector Machine

Support vector machine, which has been successfully applied to pattern recognition, regression estimation, cluster and so on, is a typical instance of kernel method. It is completely characterized by kernel function and training set. The key to enhance performance of support vector machine is to choose an appropriate kernel function for the given problem; therefore deep understanding to kernel itself is needed. Firstly, this paper analyzes some important properties of kernel, and then proposes criterions for judgment of three classes of kernel function, i.e. translation invariant, rotation invariant and convolution kernels. By them, a lot of important kernel functions are constructed some of which are commonly employed in practice.