论文标题
内核判别分析中非线性嵌入的几何形状
The Geometry of Nonlinear Embeddings in Kernel Discriminant Analysis
论文作者
论文摘要
费舍尔的线性判别分析是一种经典分类方法，但仅限于捕获线性特征。已知内核判别分析是通过非线性特征映射成功缓解限制的。我们通过识别依赖于数据分布和内核的种群级别的判别函数来研究与多项式内核和高斯内核中判别分析中非线性嵌入的几何形状。为了获得判别函数，我们通过课堂间和类协方差操作员解决了广义的特征值问题。显示多项式判别因子可通过人口矩明确地捕获类差异。为了近似高斯判别物，我们通过利用Hermite多项式的指数生成函数来使用高斯内核的特定表示。我们还表明，可以使用数据的随机投影近似高斯判别物。我们的结果阐明了数据分布和内核如何在确定非线性嵌入以进行歧视的情况下相互作用，并为选择内核及其参数提供了指南。
Fisher's linear discriminant analysis is a classical method for classification, yet it is limited to capturing linear features only. Kernel discriminant analysis as an extension is known to successfully alleviate the limitation through a nonlinear feature mapping. We study the geometry of nonlinear embeddings in discriminant analysis with polynomial kernels and Gaussian kernel by identifying the population-level discriminant function that depends on the data distribution and the kernel. In order to obtain the discriminant function, we solve a generalized eigenvalue problem with between-class and within-class covariance operators. The polynomial discriminants are shown to capture the class difference through the population moments explicitly. For approximation of the Gaussian discriminant, we use a particular representation of the Gaussian kernel by utilizing the exponential generating function for Hermite polynomials. We also show that the Gaussian discriminant can be approximated using randomized projections of the data. Our results illuminate how the data distribution and the kernel interact in determination of the nonlinear embedding for discrimination, and provide a guideline for choice of the kernel and its parameters.
