The kernel matrix used in kernel methods encodes all the information required for solving complex nonlinear problems defined on data representations in the input space using simple, but implicitly defined, solutions. Spectral analysis on the kernel matrix defines an explicit nonlinear mapping of the input data representations to a subspace of the kernel space, which can be used for directly applying linear methods. However, the selection of the kernel subspace is crucial for the performance of the proceeding processing steps. In this paper, we propose a component analysis method for kernel-based dimensionality reduction that optimally preserves the pair-wise distances of the class means in the feature space. We provide extensive analysis on the connection of the proposed criterion to those used in kernel principal component analysis and kernel discriminant analysis, leading to a discriminant analysis version of the proposed method. Our analysis also provides more insights on the properties of the feature spaces obtained by applying these methods.
翻译:内核方法中使用的内核矩阵用简单但含蓄的解决方案编码了解决输入空间中数据表示中界定的复杂非线性问题所需的所有信息。内核矩阵的光谱分析定义了内核空间子空间输入数据表示的清晰非线性绘图,可用于直接应用线性方法。然而,内核子空间的选择对于程序处理步骤的运行至关重要。在本文件中,我们提出了一个内核基维度减少的组件分析方法,以优化地物空间中分类手段的双向距离。我们就拟议标准与内核主要组成部分分析和内核共振分析中使用的标准之间的联系提供了广泛的分析,从而导致对拟议方法的辨别式分析版本。我们的分析还就通过应用这些方法而获得的特征空间的特性提供了更多的见解。