Spectral Connectivity Analysis

Spectral kernel methods are techniques for mapping data into a coordinate system that efficiently reveals the geometric structure—in particular, the “connectivity”—of the data. These methods depend on tuning parameters. We analyze the dependence of the method on these tuning parameters. We focus on one particular technique—diffusion maps—but our analysis can be used for other spectral methods as well. We identify the key population quantities, we define an appropriate risk function for analyzing the estimators, and we explain how these methods relate to classical kernel smoothing. We also show that, in some cases, fast rates of convergence are possible even in high dimensions. The Appendix of the article is available online as supplementary materials.