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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. 

Merge trees are a valuable tool in the scientific visualization of scalar fields; however, current methods for merge tree comparisons are computationally expensive, primarily due to the exhaustive matching between tree nodes. To address this challenge, we introduce the Merge Tree Neural Network (MTNN), a learned neural network model designed for merge tree comparison. The MTNN enables rapid and high-quality similarity computation. We first demonstrate how to train graph neural networks, which emerged as effective encoders for graphs, in order to to produce embeddings of merge trees in vector spaces for efficient similarity comparison. Next, we formulate the novel MTNN model that further improves the similarity comparisons by integrating the tree and node embeddings with a new topological attention mechanism. We demonstrate the effectiveness of our model on real-world data in different domains and examine our model's generalizability across various datasets. Our experimental analysis demonstrates our approach's superiority in accuracy and efficiency. In particular, we speed up the prior state-of-the-art by more than 100x on the benchmark datasets while maintaining an error rate below 0.1%. 

Recent developments in shape reconstruction and comparison call for the use of many different (topological) descriptor types, such as persistence diagrams and Euler characteristic functions. We establish a framework to quantitatively compare the strength of different descriptor types, setting up a theory that allows for future comparisons and analysis of descriptor types and that can inform choices made in applications. We use this framework to partially order a set of six common descriptor types. We then give lower bounds on the size of sets of descriptors that uniquely correspond to simplicial complexes, giving insight into the advantages of using verbose rather than concise topological descriptors. 

Recent developments in shape reconstruction and comparison call for the use of many different types of topological descriptors (persistence diagrams, Euler characteristic functions, etc.). We establish a framework that allows for quantitative comparisons of topological descriptor types and therefore may be used as a tool in more rigorously justifying choices made in applications. We then use this framework to partially order a set of six common topological descriptor types. In particular, the resulting poset gives insight into the advantages of using verbose rather than concise topological descriptors. We then provide lower bounds on the size of sets of descriptors that are complete discrete invariants of simplicial complexes, both tight and worst case. This work sets up a rigorous theory that allows for future comparisons and analysis of topological descriptor types. 

The combinatorial interpretation of the persistence diagram as a Möbius inversion was recently shown to be functorial. We employ this discovery to recast the Persistent Homology Transform of a geometric complex as a representation of a cellulation on the n-sphere to the category of combinatorial persistence diagrams. Detailed examples are provided. We hope this recasting of the PH transform will allow for the adoption of existing methods from algebraic and topological combinatorics to the study of shapes. 

This paper presents the first approach to visualize the importance of topological features that define classes of data. Topological features, with their ability to abstract the fundamental structure of complex data, are an integral component of visualization and analysis pipelines. Although not all topological features present in data are of equal importance. To date, the default definition of feature importance is often assumed and fixed. This work shows how proven explainable deep learning approaches can be adapted for use in topological classification. In doing so, it provides the first technique that illuminates what topological structures are important in each dataset in regards to their class label. In particular, the approach uses a learned metric classifier with a density estimator of the points of a persistence diagram as input. This metric learns how to reweigh this density such that classification accuracy is high. By extracting this weight, an importance field on persistent point density can be created. This provides an intuitive representation of persistence point importance that can be used to drive new visualizations. This work provides two examples: Visualization on each diagram directly and, in the case of sublevel set filtrations on images, directly on the images themselves. This work highlights real-world examples of this approach visualizing the important topological features in graph, 3D shape, and medical image data. 

Let γ be a generic closed curve in the plane. Samuel Blank, in his 1967 Ph.D. thesis, determined if γ is self-overlapping by geometrically constructing a combinatorial word from γ. More recently, Zipei Nie, in an unpublished manuscript, computed the minimum homotopy area of γ by constructing a combinatorial word algebraically. We provide a unified framework for working with both words and determine the settings under which Blank’s word and Nie’s word are equivalent. Using this equivalence, we give a new geometric proof for the correctness of Nie’s algorithm. Unlike previous work, our proof is constructive which allows us to naturally compute the actual homotopy that realizes the minimum area. Furthermore, we contribute to the theory of self-overlapping curves by providing the first polynomial-time algorithm to compute a self-overlapping decomposition of any closed curve γ with minimum area. 

The Fréchet distance is often used to measure distances between paths, with applications in areas ranging from map matching to GPS trajectory analysis to hand- writing recognition. More recently, the Fréchet distance has been generalized to a distance between two copies of the same graph embedded or immersed in a metric space; this more general setting opens up a wide range of more complex applications in graph analysis. In this paper, we initiate a study of some of the fundamental topological properties of spaces of paths and of graphs mapped to R^n under the Fréchet distance, in an eort to lay the theoretical groundwork for understanding how these distances can be used in practice. In particular, we prove whether or not these spaces, and the metric balls therein, are path-connected. 

Taking length into consideration while comparing 1D shapes is a challenging task. In particular, matching equal-length portions of such shapes regardless of their combinatorial features, and only based on proximity, is often required in biomedical and geospatial applications. In this work, we define the length-sensitive partial Fréchet similarity (LSFS) between curves (or graphs), which maximizes the length of matched portions that are close to each other and of equal length. We present an exact polynomial-time algorithm to compute LSFS between curves under and . For geometric graphs, we show that the decision problem is NP-hard even if one of the graphs consists of one edge.