Spectral Clustering and Kernel PCA are Learning Eigenfunctions

In this paper, we show a direct equivalence between spectral clustering and kernel PCA, and how both are special cases of a more general learning problem, that of learning the principal eigenfunctions of a kernel, when the functions are from a function space whose scalar product is defined with respect to a density model. This defines a natural mapping for new data points, for methods that only provided an embedding, such as spectral clustering and Laplacian eigenmaps. The analysis hinges on a notion of generalization for embedding algorithms based on the estimation of underlying eigenfunctions, and suggests ways to improve this generalization by smoothing the data empirical distribution.