Diffusion maps for changing data

TL;DR

This paper introduces a method to analyze how data embeddings evolve with changing parameters using diffusion maps, enabling tracking of intrinsic geometric changes and defining a global diffusion distance.

Contribution

It provides a framework for mapping and comparing diffusion embeddings across parameter variations, including approximation theorems and potential applications.

Findings

Method for transitioning between diffusion embeddings

Definition of a global diffusion distance

Approximation theorems for sampled data

Abstract

Graph Laplacians and related nonlinear mappings into low dimensional spaces have been shown to be powerful tools for organizing high dimensional data. Here we consider a data set X in which the graph associated with it changes depending on some set of parameters. We analyze this type of data in terms of the diffusion distance and the corresponding diffusion map. As the data changes over the parameter space, the low dimensional embedding changes as well. We give a way to go between these embeddings, and furthermore, map them all into a common space, allowing one to track the evolution of X in its intrinsic geometry. A global diffusion distance is also defined, which gives a measure of the global behavior of the data over the parameter space. Approximation theorems in terms of randomly sampled data are presented, as are potential applications.