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1. Measure-Theoretic Reeb Graphs and Reeb Spaces

   https://doi.org/10.4230/LIPIcs.SoCG.2024.80  

   Wang, Qingsong; Ma, Guanqun; Sridharamurthy, Raghavendra; Wang, Bei (June 2024, 40th International Symposium on Computational Geometry (SoCG 2024), Leibniz International Proceedings in Informatics (LIPIcs))

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   A Reeb graph is a graphical representation of a scalar function on a topological space that encodes the topology of the level sets. A Reeb space is a generalization of the Reeb graph to a multiparameter function. In this paper, we propose novel constructions of Reeb graphs and Reeb spaces that incorporate the use of a measure. Specifically, we introduce measure-theoretic Reeb graphs and Reeb spaces when the domain or the range is modeled as a metric measure space (i.e., a metric space equipped with a measure). Our main goal is to enhance the robustness of the Reeb graph and Reeb space in representing the topological features of a scalar field while accounting for the distribution of the measure. We first introduce a Reeb graph with local smoothing and prove its stability with respect to the interleaving distance. We then prove the stability of a Reeb graph of a metric measure space with respect to the measure, defined using the distance to a measure or the kernel distance to a measure, respectively. 

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2. A Family of Metrics from the Truncated Smoothing of Reeb Graphs

   https://doi.org/10.4230/LIPIcs.SoCG.2021.22  

   Chambers, Erin Wolf; Munch, Elizabeth; Ophelders, Tim (June 2021, Forensic toxicology)

   Buchin, Kevin; Colin de Verdi\` (Ed.)

   In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we "chop off" parts near local minima and maxima during the course of smoothing, where the amount cut is controlled by a parameter τ. After formalizing truncation as a functor, we show that when applied after the smoothing functor, this prevents extensive expansion of the range of the function, and yields particularly nice properties (such as maintaining connectivity) when combined with smoothing for 0 ≤ τ ≤ 2ε, where ε is the smoothing parameter. Then, for the restriction of τ ∈ [0,ε], we have additional structure which we can take advantage of to construct a categorical flow for any choice of slope m ∈ [0,1]. Using the infrastructure built for a category with a flow, this then gives an interleaving distance for every m ∈ [0,1], which is a generalization of the original interleaving distance, which is the case m = 0. While the resulting metrics are not stable, we show that any pair of these for m, m' ∈ [0,1) are strongly equivalent metrics, which in turn gives stability of each metric up to a multiplicative constant. We conclude by discussing implications of this metric within the broader family of metrics for Reeb graphs. 

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3. Stability and Approximations for Decorated Reeb Spaces

   https://doi.org/10.4230/LIPIcs.SoCG.2024.44  

   Curry, Justin; Mio, Washington; Needham, Tom; Okutan, Osman Berat; Russold, Florian (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   Given a map f:X → M from a topological space X to a metric space M, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when M = ℝ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of f. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data. 

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4. Computing Reeb dynamics on four-dimensional convex polytopes

   https://doi.org/10.3934/jcd.2021016  

   Chaidez, Julian; Hutchings, Michael (January 2021, Journal of Computational Dynamics)

   null (Ed.)

   We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio \begin{document}$ 1 $$\end{document}$. 

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5. Learning Simplicial Complexes from Persistence Diagrams

   Belton, Robin Lynne; Fasy, Brittany Terese; Mertz, Rostik; Micka, Samuel; Millman, David L.; Salinas, Daniel; Schenfisch, Anna; Schupbach, Jordan; Williams, Lucia (August 2018, Canadian Conference on Computational Geometry)

   Topological Data Analysis (TDA) studies the “shape” of data. A common topological descriptor is the persistence diagram, which encodes topological features in a topological space at different scales. Turner, Mukherjee, and Boyer showed that one can reconstruct a simplicial complex embedded in R^3 using persistence diagrams generated from all possible height filtrations (an uncountably infinite number of directions). In this paper, we present an algorithm for reconstructing plane graphs K = (V, E) in R^2, i.e., a planar graph with vertices in general position and a straight-line embedding, from a quadratic number height filtrations and their respective persistence diagrams. 

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