Time-inhomogeneous diffusion geometry and topology

TL;DR

This paper provides a comprehensive theoretical analysis of diffusion condensation, a process for multiscale data representation, exploring its convergence, geometric, spectral, and topological properties, and connecting it to hierarchical clustering and topological data analysis.

Contribution

It introduces new convergence bounds and topological insights for diffusion condensation, extending understanding beyond homogeneous processes and linking it to topological data analysis.

Findings

Convergence bounds depend on transition probability and data radius.

Spectral bounds relate to the eigenspectrum of the diffusion kernel.

Diffusion condensation generalizes hierarchical clustering and defines an intrinsic topology.

Abstract

Diffusion condensation is a dynamic process that yields a sequence of multiscale data representations that aim to encode meaningful abstractions. It has proven effective for manifold learning, denoising, clustering, and visualization of high-dimensional data. Diffusion condensation is constructed as a time-inhomogeneous process where each step first computes and then applies a diffusion operator to the data. We theoretically analyze the convergence and evolution of this process from geometric, spectral, and topological perspectives. From a geometric perspective, we obtain convergence bounds based on the smallest transition probability and the radius of the data, whereas from a spectral perspective, our bounds are based on the eigenspectrum of the diffusion kernel. Our spectral results are of particular interest since most of the literature on data diffusion is focused on homogeneous processes. From a topological perspective, we show diffusion condensation generalizes centroid-based hierarchical clustering. We use this perspective to obtain a bound based on the number of data points, independent of their location. To understand the evolution of the data geometry beyond convergence, we use topological data analysis. We show that the condensation process itself defines an intrinsic condensation homology. We use this intrinsic topology as well as the ambient persistent homology of the condensation process to study how the data changes over diffusion time. We demonstrate both types of topological information in well-understood toy examples. Our work gives theoretical insights into the convergence of diffusion condensation, and shows that it provides a link between topological and geometric data analysis.