Convergence between Categorical Representations of Reeb Space and Mapper

TL;DR

This paper establishes a formal theoretical link between the Reeb space and the Mapper construction, proving their convergence using category theory and interleaving distance, thus supporting their use in topological data analysis.

Contribution

It provides the first formal proof of convergence between Mapper and Reeb space, quantifying approximation quality via interleaving distance.

Findings

Proves Mapper converges to Reeb space in the limit.

Quantifies approximation quality at fixed resolutions.

Uses category theory to formalize the relationship.

Abstract

The Reeb space, which generalizes the notion of a Reeb graph, is one of the few tools in topological data analysis and visualization suitable for the study of multivariate scientific datasets. First introduced by Edelsbrunner et al., it compresses the components of the level sets of a multivariate mapping and obtains a summary representation of their relationships. A related construction called mapper, and a special case of the mapper construction called the Joint Contour Net have been shown to be effective in visual analytics. Mapper and JCN are intuitively regarded as discrete approximations of the Reeb space, however without formal proofs or approximation guarantees. An open question has been proposed by Dey et al. as to whether the mapper construction converges to the Reeb space in the limit. In this paper, we are interested in developing the theoretical understanding of the relationship between the Reeb space and its discrete approximations to support its use in practical data analysis. Using tools from category theory, we formally prove the convergence between the Reeb space and mapper in terms of an interleaving distance between their categorical representations. Given a sequence of refined discretizations, we prove that these approximations converge to the Reeb space in the interleaving distance; this also helps to quantify the approximation quality of the discretization at a fixed resolution.