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. We propose a new optimization criterion, leading to a new component analysis method for kernel-based dimensionality reduction that optimally preserves the pair-wise distances of the class means in the feature space. This leads to efficient kernel subspace learning, which is crucial for kernel-based machine learning solutions. We provide extensive analysis on the connections and differences between the proposed criterion and the criteria used in kernel Principal Component Analysis, kernel Entropy Component Analysis and Kernel Discriminant Analysis, leading to a discriminant analysis version of the proposed method. Our theoretical analysis also provides more insights on the properties of the feature spaces obtained by applying these methods. Results on a variety of visual classification problems illustrate the properties of the proposed methods.

内核方法中使用的内核矩阵使用简单但隐式定义的解决方案对解决在输入空间中的数据表示形式上定义的复杂非线性问题所需的所有信息进行编码。对内核矩阵的频谱分析定义了输入数据表示形式到内核空间子空间的显式非线性映射，可用于直接应用线性方法。但是，内核子空间的选择对于后续处理步骤的执行至关重要。我们提出了一种新的优化准则，从而为基于核的降维提供了一种新的成分分析方法，该方法可以最佳地保留特征空间中类均值的成对距离。这导致有效的内核子空间学习，这对于基于内核的机器学习解决方案至关重要。我们对提出的标准与内核主成分分析，内核熵成分分析和内核判别分析中使用的标准之间的联系和差异进行了广泛的分析，从而得出了该方法的判别分析版本。我们的理论分析还提供了有关通过应用这些方法获得的特征空间的属性的更多见解。各种视觉分类问题的结果说明了所提出方法的性质。我们的理论分析还提供了有关通过应用这些方法获得的特征空间的属性的更多见解。各种视觉分类问题的结果说明了所提出方法的性质。我们的理论分析还提供了有关通过应用这些方法获得的特征空间的属性的更多见解。各种视觉分类问题的结果说明了所提出方法的性质。