Reeb Space Approximation with Guarantees

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. [3], the Reeb space of a multivariate mapping f : X→ R parameterizes the set of components of preimages of points in R. In this paper, we formally prove the convergence between the Reeb space and its discrete approximation, mapper, in terms of an interleaving distance between them. At a fixed resolution of the discretization, such a distance allows us to quantify the approximation quality and leads to a Reeb space approximation algorithm with guarantees based upon established techniques. 1 Reeb Graphs and Reeb Spaces Multivariate datasets arise in many scientific applications, ranging from oceanography to astrophysics, from chemistry to meteorology, from nuclear engineering to molecular dynamics. Consider, for example, combustion or climate simulation where various physical measurements (e.g. temperature and pressure) or concentrations of multiple chemical species are computed simultaneously. We model these variables mathematically as multiple continuous, realvalued functions defined on a shared domain and represented as a multivariate mapping f = {f1, · · · , fr} : X → R. We are interested in understanding the relationships between these real-valued functions, and more generally, in developing efficient and effective tools for their analysis and visualization. When r = 1, we can study the Reeb graph [6], which contracts each contour (i.e. component of a level set) of a real-valued function to a single point and uses a graph representation to summarize the connections between these contours. When the domain is simply connected, this construction is referred to as a contour tree [8]. Recent work by de Silva et al. [2] has shown that the data of a Reeb graph can be stored in a category-theoretic object called a cosheaf, which opened the way for defining a metric for Reeb graphs known as the interleaving distance. In the case of multivariate data (r ≥ 1), we generalize the idea of the Reeb graph to the Reeb space [3]. Let f : X → R be a generic, continuous, piecewise linear (PL) mapping defined on a combinatorial d-manifold. Intuitively, the Reeb space of f parametrizes the set of components of preimages of points in R [3]. Two points x, y ∈ X are equivalent, denoted by x ∼ y, if f(x) = f(y) and x and y belong to the same path connected component of the preimage, f−1(f(x)) = f−1(f(y)). The Reeb space is the quotient space obtained by identifying equivalent points, that is, RSf = X/ ∼, together with the quotient topology inherited from X. The Reeb graph can then be considered a special case in this context when r = 1. Reeb spaces have been shown to have triangulations and canonical stratifications into manifolds [3].