Spectral Connectivity Analysis

TL;DR

This paper analyzes spectral kernel methods, especially diffusion maps, focusing on their dependence on tuning parameters, their relation to kernel smoothing, and their convergence properties in high-dimensional data.

Contribution

It provides a theoretical analysis of spectral kernel methods, clarifying their implicit population quantities and establishing convergence rates even in high-dimensional settings.

Findings

Identification of population quantities estimated by spectral methods

Relation of spectral methods to classical kernel smoothing

Possibility of fast convergence rates in high dimensions

Abstract

Spectral kernel methods are techniques for transforming data into a coordinate system that efficiently reveals the geometric structure - in particular, the "connectivity" - of the data. These methods depend on certain 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 methods as well. We identify the population quantities implicitly being estimated, we explain how these methods relate to classical kernel smoothing and we define an appropriate risk function for analyzing the estimators. We also show that, in some cases, fast rates of convergence are possible even in high dimensions.