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arXiv · 2312.01982

Stability and Approximations for Decorated Reeb Spaces

Justin Curry·Washington Mio·Tom Needham·Osman Berat Okutan·Florian Russold
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Abstract

Given a map $f:X \to M$ 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 $M=\mathbb{R}$ 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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Justin Curry, Washington Mio, Tom Needham, Osman Berat Okutan, Florian Russold. 2023-12-04. Stability and Approximations for Decorated Reeb Spaces. https://arxiv.org/abs/2312.01982

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