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  1. Abstract

    Graphs in metric spaces appear in a wide range of data sets, and there is a large body of work focused on comparing, matching, or analyzing collections of graphs in different ambient spaces. In this survey, we provide an overview of a diverse collection of distance measures that can be defined on the set of finite graphs immersed (and in some cases, embedded) in a metric space. For each of the distance measures, we recall their definitions and investigate which of the properties of a metric they satisfy. Furthermore we compare the distance measures based on these properties and discuss their computational complexity.

     
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  2. The weighted Euler curve transform (WECT) was recently introduced as a tool to extract meaningful information from shape data, when the shape is equipped with a weight function. In this extended abstract, we provide an experimental investigation on using the WECT for image classification. 
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    Free, publicly-accessible full text available June 12, 2024
  3. Since its introduction in the mid-1990s, DBSCAN has become one of the most widely used clustering algorithms. However, one of the steps in DBSCAN is to perform a range query, a task that is difficult in many spaces, including the space of persistence diagrams. In this paper, we introduce a spanner into the DBSCAN algorithm to facilitate range queries in such spaces. We provide a proof-of-concept implementation, and study time and clustering performance for two data sets of persistence diagrams. 
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  4. We examine topological properties of spaces of paths and graphs mapped to $\R^d$ under the Fr\'echet distance. We show that these spaces are path-connected if the map is either continuous or an immersion. If the map is an embedding, we show that the space of paths is path-connected, while the space of graphs only maintains this property in dimensions four or higher. 
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  5. We consider the topological and geometric reconstruction of a geodesic subspace of [Formula: see text] both from the Čech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages the intrinsic length metric induced by the geodesics on the subspace. We consider the distortion and convexity radius as our sampling parameters for the reconstruction problem. For a geodesic subspace with finite distortion and positive convexity radius, we guarantee a correct computation of its homotopy and homology groups from the sample. This technique provides alternative sampling conditions to the existing and commonly used conditions based on weak feature size and [Formula: see text]–reach, and performs better under certain types of perturbations of the geodesic subspace. For geodesic subspaces of [Formula: see text], we also devise an algorithm to output a homotopy equivalent geometric complex that has a very small Hausdorff distance to the unknown underlying space. 
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  6. We derive conditions under which the reconstruction of a target space is topologically correct via the Čech complex or the Vietoris-Rips complex obtained from possibly noisy point cloud data. We provide two novel theoretical results. First, we describe sufficient conditions under which any non-empty intersection of finitely many Euclidean balls intersected with a positive reach set is contractible, so that the Nerve theorem applies for the restricted Čech complex. Second, we demonstrate the homotopy equivalence of a positive μ-reach set and its offsets. Applying these results to the restricted Čech complex and using the interleaving relations with the Čech complex (or the Vietoris-Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the Čech complex (or the Vietoris-Rips complex), in terms of the μ-reach. Our results sharpen existing results. 
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  7. Given a simplicial complex K and an injective function f from the vertices of K to R, we consider algorithms that extend f to a discrete Morse function on K. We show that an algorithm of King, Knudson and Mramor can be described on the directed Hasse diagram of K. Our description has a faster runtime for high dimensional data with no increase in space. 
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