田萌, 王文剑. 高斯核函数选择的广义核极化准则[J]. 计算机研究与发展, 2015, 52(8): 1722-1734.
 引用本文: 田萌, 王文剑. 高斯核函数选择的广义核极化准则[J]. 计算机研究与发展, 2015, 52(8): 1722-1734.
Tian Meng, Wang Wenjian. Generalized Kernel Polarization Criterion for Optimizing Gaussian Kernel[J]. Journal of Computer Research and Development, 2015, 52(8): 1722-1734.
 Citation: Tian Meng, Wang Wenjian. Generalized Kernel Polarization Criterion for Optimizing Gaussian Kernel[J]. Journal of Computer Research and Development, 2015, 52(8): 1722-1734.

## Generalized Kernel Polarization Criterion for Optimizing Gaussian Kernel

• 摘要: 核函数及其参数的选择是核方法研究中的一个基本却很困难的问题,高斯核是目前各类核方法中最常使用的一种核函数.关于高斯核参数的优化已有很多研究，然而这些方法大多存在时间复杂度高，或是算法实现困难，或是样本数据需服从多元正态分布的前提假设等不足.提出的广义核极化准则可用来解决分类问题中的高斯核参数优化，该准则通过保持类内局部结构信息及中心化核矩阵以更准确地刻画特征空间中类别间的分离度，进而获得更好的高斯核参数来提高分类性能.给出了广义核极化准则对应目标函数的近似最优解的存在唯一性证明，且由于该准则独立于学习算法，因此可用许多成熟的优化算法来寻找最优参数.此外，还补充了已有文献提出的局部核极化准则对应目标函数近似最优解的存在唯一性证明，并且指出该准则是所提出的广义核极化准则的一个特例.针对多分类问题，分别给出广义核极化准则及局部核极化准则的多分类拓展形式.在标准数据集上的实验结果表明所提准则的有效性.

Abstract: The choice of kernel function is a basic and challenging problem in researches on kernel methods. Gaussian kernel is a popular and widely used one in various kernel methods, and many universal kernel selection methods have been derived for Gaussian kernel. However, these methods may have some disadvantages, such as heavy computational complexity, the difficulty of algorithm implement, and the requirement of the classes generated from underlying multivariate normal distributions. To remedy these problems, generalized kernel polarization criterion has been proposed to tune the parameter of Gaussian kernel for classification tasks. By taking the within-class local structure into account and centering the kernel matrix, the criterion does better in maximizing the class separability in the feature space. And the final optimized kernel parameter leads to a substantial improvement in the performance. Furthermore, the criterion function can be proved to have a determined approximate global minimum point. This good characteristic, coupled with its independence of the actual learning machine, makes the optimal parameter easier to find by many algorithms. Besides this, local kernel polarization criterion function, a special case of generalized kernel polarization criterion function, can also be proved to have a determined approximate global minimum point. The extensions of generalized kernel polarization criterion and local kernel polarization criterion to the multiclass domain have been proposed. Experimental results show the effectiveness and efficiency of our proposed criteria.

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