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1. Maximum Subbarcode Matching and Subbarcode Distance

   Chubet, Oliver (January 2022, Canadian Conference in Computational Geometry)

   Bahoo, Yeganeh; Georgiou, Konstantinos (Ed.)

   We investigate the maximum subbarcode matching problem which arises from the study of persistent homology and introduce the subbarcode distance on barcodes. A barcode is a set of intervals which correspond to topological features in data and is the output of a persistent homology computation. A barcode A has a subbarcode matching to B if each interval in A matches to an interval in B which contains it. We present an algorithm which takes two barcodes, A and B, and returns a maximum subbarcode matching. 

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2. A topological study of functional data and Frechet functions of metric measure spaces

   https://doi.org/10.1007/s41468-019-00037-8  

   Hang, H; Memoli, F; Mio, W. (August 2019, Journal of applied and computational topology)

   We study the persistent homology of both functional data on compact topological spaces and structural data presented as compact metric measure spaces. One of our goals is to define persistent homology so as to capture primarily properties of the shape of a signal, eliminating otherwise highly persistent homology classes that may exist simply because of the nature of the domain on which the signal is defined. We investigate the stability of these invariants using metrics that downplay regions where signals are weak. The distance between two signals is small if they exhibit high similarity in regions where they are strong, regardless of the nature of their full domains, in particular allowing different homotopy types. Consistency and estimation of persistent homology of metric measure spaces from data are studied within this framework. We also apply the methodology to the construction of multi-scale topological descriptors for data on compact Riemannian manifolds via metric relaxations derived from the heat kernel. 

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3. Topological analysis of traffic pace via persistent homology*

   https://doi.org/10.1088/2632-072X/abc96a  

   Carmody, Daniel R.; Sowers, Richard B. (January 2021, Journal of Physics: Complexity)

   Abstract We develop a topological analysis of robust traffic pace patterns using persistent homology. We develop Rips filtrations, parametrized by pace, for a symmetrization of traffic pace along the (naturally) directed edges in a road network. Our symmetrization is inspired by recent work of Turner (2019Algebr. Geom. Topol.191135–1170). Our goal is to construct barcodes which help identify meaningful pace structures, namely connected components or ‘rings’. We develop a case study of our methods using datasets of Manhattan and Chengdu traffic speeds. In order to cope with the computational complexity of these large datasets, we develop an auxiliary application of the directed Louvain neighborhood-finding algorithm. We implement this as a preprocessing step prior to our main persistent homology analysis in order to coarse-grain small topological structures. We finally compute persistence barcodes on these neighborhoods. The persistence barcodes have a metric structure which allows us to both qualitatively and quantitatively compare traffic networks. As an example of the results, we find robust connected pace structures near Midtown bridges connecting Manhattan to the mainland. 

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4. Topological data analysis of human brain networks through order statistics

   https://doi.org/10.1371/journal.pone.0276419  

   Das, Soumya; Anand, D. Vijay; Chung, Moo K. (March 2023, PLOS ONE)

   Topaz, Chad M. (Ed.)

   Understanding the common topological characteristics of the human brain network across a population is central to understanding brain functions. The abstraction of human connectome as a graph has been pivotal in gaining insights on the topological properties of the brain network. The development of group-level statistical inference procedures in brain graphs while accounting for the heterogeneity and randomness still remains a difficult task. In this study, we develop a robust statistical framework based on persistent homology using theorder statisticsfor analyzing brain networks. The use of order statistics greatly simplifies the computation of the persistent barcodes. We validate the proposed methods using comprehensive simulation studies and subsequently apply to the resting-state functional magnetic resonance images. We found a statistically significanttopologicaldifference between the male and female brain networks. 

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5. Ephemeral persistence features and the stability of filtered chain complexes

   https://doi.org/10.20382/jocg.v15i2a8  

   Zhou, Ling; Mémoli, Facundo (January 2025, Journal of computational geometry)

   We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of filtered chain complexes. The information present at the level of filtered chain complexes includes points with zero persistence which provide additional information to that present at homology level. The resulting invariant, called verbose barcode, which has a stronger discriminating power than the usual barcode, is proved to be stable under certain metrics that are sensitive to these ephemeral points. In some situations, we provide ways to compute such metrics between verbose barcodes. We also exhibit several examples of finite metric spaces with identical (standard) VR barcodes yet with different verbose VR barcodes, thus confirming that these ephemeral points strengthen the standard VR barcode. 

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