On Feature Extraction via Kernels

Using the kernel trick idea and the kernels-as-features idea, we can construct two kinds of nonlinear feature spaces, where linear feature extraction algorithms can be employed to extract nonlinear features. In this correspondence, we study the relationship between the two kernel ideas applied to certain feature extraction algorithms such as linear discriminant analysis, principal component analysis, and canonical correlation analysis. We provide a rigorous theoretical analysis and show that they are equivalent up to different scalings on each feature. These results provide a better understanding of the kernel method.