Vietoris-Rips and Cech Complexes of Metric Gluings
Michał AdamaszekHenry AdamsEllen GasparovicMaria GommelEmilie PurvineRadmila SazdanovićBei WangYusu WangLori Ziegelmeier
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Abstract:
We study Vietoris-Rips and Cech complexes of metric wedge sums and metric gluings. We show that the Vietoris-Rips (resp. Cech) complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris-Rips (resp. Cech) complexes. We also provide generalizations for certain metric gluings, i.e. when two metric spaces are glued together along a common isometric subset. As our main example, we deduce the homotopy type of the Vietoris-Rips complex of two metric graphs glued together along a sufficiently short path. As a result, we can describe the persistent homology, in all homological dimensions, of the Vietoris-Rips complexes of a wide class of metric graphs.Keywords:
Wedge (geometry)
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Flag (linear algebra)
Simplicial approximation theorem
h-vector
Abstract simplicial complex
Simplicial homology
Simplex
Algebraic topology
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We study Vietoris-Rips and Cech complexes of metric wedge sums and metric gluings. We show that the Vietoris-Rips (resp. Cech) complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris-Rips (resp. Cech) complexes. We also provide generalizations for certain metric gluings, i.e. when two metric spaces are glued together along a common isometric subset. As our main example, we deduce the homotopy type of the Vietoris-Rips complex of two metric graphs glued together along a sufficiently short path. As a result, we can describe the persistent homology, in all homological dimensions, of the Vietoris-Rips complexes of a wide class of metric graphs.
Wedge (geometry)
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Surjective function
Simplicial approximation theorem
Abstract simplicial complex
Finite set
Simplicial homology
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To any finite simplicial complex X, we associate a natural filtration starting from Chari and Joswig's discrete Morse complex and abutting to the matching complex of X. This construction leads to the definition of several homology theories, which we compute in a number of examples. We also completely determine the graded object associated to this filtration in terms of the homology of simpler complexes. This last result provides some connections to the number of vertex-disjoint cycles of a graph.
Simplicial homology
Disjoint sets
Homology
Simplicial approximation theorem
Filtration (mathematics)
Abstract simplicial complex
Discrete Morse theory
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We introduce a new invariant defined on the vertices of a given filtered simplicial complex, called codensity, which controls the impact of removing vertices on persistent homology. We achieve this control through the use of an interleaving type of distance between fitered simplicial complexes. We study the special case of Vietoris-Rips filtrations and show that our bounds offer a significant improvement over the immediate bounds coming from considerations related to the Gromov-Hausdorff distance. Based on these ideas we give an iterative method for the practical simplification of filtered simplicial complexes.
As a byproduct of our analysis we identify a notion of core of a filtered simplicial complex which admits the interpretation as a minimalistic simplicial filtration which retains all the persistent homology information.
Simplicial homology
h-vector
Abstract simplicial complex
Simplicial approximation theorem
Simplicial manifold
Homology
Persistent Homology
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Ellipse
Eccentricity (behavior)
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We study the simplification of simplicial complexes by repeated edge contractions. First, we extend to arbitrary simplicial complexes the statement that edges satisfying the link condition can be contracted while preserving the homotopy type. Our primary interest is to simplify flag complexes such as Rips complexes for which it was proved recently that they can provide topologically correct reconstructions of shapes. Flag complexes (sometimes called clique complexes) enjoy the nice property of being completely determined by the graph of their edges. But, as we simplify a flag complex by repeated edge contractions, the property that it is a flag complex is likely to be lost. Our second contribution is to propose a new representation for simplicial complexes particularly well adapted for complexes close to flag complexes. The idea is to encode a simplicial complex K by the graph G of its edges together with the inclusion-minimal simplices in the set difference G - K. We call these minimal simplices blockers. We prove that the link condition translates nicely in terms of blockers and give formulae for updating our data structure after an edge contraction. Finally, we observe in some simple cases that few blockers appear during the simplification of Rips complexes, demonstrating the efficiency of our representation in this context.
Flag (linear algebra)
Simplicial approximation theorem
Abstract simplicial complex
Simplicial homology
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Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial complex itself. The difference between the simplicial complex and such an approximation is quantitatively measured by means of the so called obstruction complexes. Our general machinery is then specialized to clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric gluings. For the latter we give metric conditions which allow to recover the first and zero-th homology of the gluing from the respective homologies of the components.
Simplicial homology
Simplicial approximation theorem
Simplicial manifold
Persistent Homology
Homology
Abstract simplicial complex
h-vector
Model category
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