Categorified Reeb Graphs

@article{Silva2016CategorifiedRG,
  title={Categorified Reeb Graphs},
  author={Vin de Silva and E. Munch and Amit K. Patel},
  journal={Discrete \& Computational Geometry},
  year={2016},
  volume={55},
  pages={854-906}
}
The Reeb graph is a construction which originated in Morse theory to study a real-valued function defined on a topological space. More recently, it has been used in various applications to study noisy data which creates a desire to define a measure of similarity between these structures. Here, we exploit the fact that the category of Reeb graphs is equivalent to the category of a particular class of cosheaf. Using this equivalency, we can define an ‘interleaving’ distance between Reeb graphs… 
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106 Citations
Highly Influential Citations
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106 Citations

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Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs
TLDR
This paper shows that the two metrics are strongly equivalent on the space of Reeb graphs, and gives an immediate proof of bottleneck stability for persistence diagrams in terms of the Reeb graph interleaving distance.
Combinatorial models for topological Reeb spaces
There are two rather distinct approaches to Morse theory nowadays: smooth and discrete. We propose to study a real valued function by assembling all associated sections in a topological category.…
Reeb Graphs of Piecewise Linear Functions
TLDR
This paper is devoted to provide a formal guarantee of the fact that the output of the transform of the Reeb graph is a graph.
Realizable piecewise linear paths of persistence diagrams with Reeb graphs
TLDR
This paper investigates the effect on the persistence diagram of a particular continuous operation on Reeb graphs; namely the (truncated) smoothing operation, and shows which paths in persistence diagram space (commonly known as vineyards) can be realized by a path in the space of Reeb graph via these simple operations.
The Reeb Graph Edit Distance is Universal
TLDR
An edit distance for Reeb graphs is defined and it is proved that it is stable and universal, meaning that it provides an upper bound to any other stable distance.
The Reeb Graph Edit Distance is Universal
TLDR
An edit distance for Reeb graphs is defined and it is proved that it is stable and universal, meaning that it provides an upper bound to any other stable distance.
Reeb complexes and topological persistence
We introduce Reeb complexes in order to capture how generators of homology flow along sections of a real valued continuous function. This intuition suggests a close relation of Reeb complexes to…
Tree decomposition of Reeb graphs, parametrized complexity, and applications to phylogenetics
TLDR
It is shown that, inside the larger category of partially ordered Reeb graphs, every Reeb graph with n leaves and first Betti number s can be identified with a coproduct of at most 2 s partially ordered trees with(n + s) leaves, and is classified up to isomorphism by their tree-decomposition.
Convergence between Categorical Representations of Reeb Space and Mapper
TLDR
Using tools from category theory, it is formally proved that the convergence between the Reeb space and mapper is proved in terms of an interleaving distance between their categorical representations.
Reeb Graph Metrics from the Ground Up
TLDR
This work surveys five existing metrics that have been defined on Reeb graphs: the bottleneck distance, the interleaving distance, functional distortion Distance, the Reeb graph edit distance, and the universal edit distance.
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References

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Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs
TLDR
This paper shows that the two metrics are strongly equivalent on the space of Reeb graphs, and gives an immediate proof of bottleneck stability for persistence diagrams in terms of the Reeb graph interleaving distance.
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TLDR
It is proved that filamentary structures that can be seen as topological metric graphs can be approximated, with respect to the Gromov-Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and an efficient and easy to implement algorithm to compute such approximations is provided.
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TLDR
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TLDR
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TLDR
This paper proves that metric reconstruction problem of such filamentary structures from data sampled around them can be approximated, with respect to the Gromov-Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and provides an efficient and easy to implement algorithm to compute such approximations in almost linear time.
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TLDR
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