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The coherent matching distance in 2D persistent homology [article]

Andrea Cerri, Marc Ethier, Patrizio Frosini
2016 arXiv   pre-print
In our paper we prove that the coherent 2D matching distance is well-defined and stable.  ...  In this paper we introduce a new matching distance for 2D persistent Betti numbers, called coherent matching distance and based on matchings that change coherently with the filtrations we take into account  ...  In Section 2, we introduce the coherent 2D matching distance and prove that it is well-defined and stable.  ... 
arXiv:1603.03886v1 fatcat:zqytvqkmtrfpvejdorhhodl4dq

On the geometrical properties of the coherent matching distance in 2D persistent homology [article]

Andrea Cerri, Marc Ethier, Patrizio Frosini
2018 arXiv   pre-print
In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way.  ...  As a by-product, we obtain a theoretical framework for managing the phenomenon of monodromy in 2D persistent homology.  ...  Acknowledgement Work carried out under the auspices of INdAM-GNSAGA.  ... 
arXiv:1801.06636v2 fatcat:fzmgzwk6rfcovnnpiywyviup7u

Geometry in the space of persistence modules

Vin de Silva, Vidit Nanda
2013 Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13  
We show that the relationship between thě Cech and Rips complexes is governed by certain 'coherence' conditions on the corresponding families of interleavings or matchings.  ...  The metric structures are determined in terms of interleaving maps between modules and matchings between diagrams.  ...  The authors thank Peter Bubenik and Konstantin Mischaikow for valuable discussions. The first author held a Digiteo Chair at INRIA SaclayÎle-de-France, where some of this work was carried out.  ... 
doi:10.1145/2462356.2462402 dblp:conf/compgeom/SilvaN13 fatcat:unxmuob3uvbxfobduojhqwyps4

Geometry in the space of persistence modules

Vin de Silva, Vidit Nanda
2013 Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13  
We show that the relationship between thě Cech and Rips complexes is governed by certain 'coherence' conditions on the corresponding families of interleavings or matchings.  ...  The metric structures are determined in terms of interleaving maps between modules and matchings between diagrams.  ...  The authors thank Peter Bubenik and Konstantin Mischaikow for valuable discussions. The first author held a Digiteo Chair at INRIA SaclayÎle-de-France, where some of this work was carried out.  ... 
doi:10.1145/2493132.2462402 fatcat:ihmefquhobdydntlkegbdritoy

Dynamic Topological Data Analysis for Brain Networks via Wasserstein Graph Clustering [article]

Moo K. Chung, Shih-Gu Huang, Ian C. Carroll, Vince D. Calhoun, H. Hill Goldsmith
2022 arXiv   pre-print
The method applied in more accurate determination of the state spaces of dynamically changing functional brain networks.  ...  The Wasserstein clustering is shown to outperform the widely used k-means clustering.  ...  In [15, 17] , persistent homology was shown to outperform various matrix norm based network distances.  ... 
arXiv:2201.00087v2 fatcat:4j47rap2ybgyxdk7fnfgt75vzy

A topological encoding convolutional neural network for segmentation of 3D multiphoton images of brain vasculature using persistent homology

Mohammad Haft-Javaherian, Martin Villiger, Chris B. Schaffer, Nozomi Nishimura, Polina Golland, Brett E. Bouma
2020 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)  
Here, we propose a topological encoding convolutional neural network based on persistent homology to segment 3D multiphoton images of brain vasculature.  ...  We demonstrate that our model out-performs state-of-the-art models in terms of the Dice coefficient and it is comparable in terms of other metrics such as sensitivity.  ...  Acknowledgment This work was supported by the National Institutes of Health NIBIB under Grant P41EB-015903 and Grant P41EB015902. In addtion, MH was supported by Bullock Postdoctoral Fellowship.  ... 
doi:10.1109/cvprw50498.2020.00503 pmid:33889437 pmcid:PMC8059194 dblp:conf/cvpr/Haft-Javaherian20 fatcat:ssbhaulpuza6fdbkicl7xtfxji

Capturing dynamics of time-varying data via topology

Lu Xian, Henry Adams, Chad M. Topaz, Lori Ziegelmeier
2021 Foundations of Data Science  
While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied studies have turned to data that varies in time.  ...  There is often a need to simplify or summarize the dynamic behavior.  ...  By the stability of persistent homology [17, Theorem 5.2], if X and Y are compact metric spaces, then the bottleneck distance satisfies d b (PH(VR(X)), PH(VR(Y ))) ≤ 2d GH (X, Y ).  ... 
doi:10.3934/fods.2021033 fatcat:l3i6gy74ofdcroyjourirxbfpe

Capturing Dynamics of Time-Varying Data via Topology [article]

Lu Xian, Henry Adams, Chad M. Topaz, Lori Ziegelmeier
2021 arXiv   pre-print
While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied studies have turned to data that varies in time.  ...  There is often a need to simplify or summarize the dynamic behavior.  ...  In bijection (ii), we match (3, 6) in Dgm k (V ) with (1, 7) in Dgm k (W ) and (2, 8) in Dgm k (V ) with (3, 7.5) in Dgm k (W ). The L ∞ distance between matched points is 2.  ... 
arXiv:2010.05780v2 fatcat:uc4fie4nwzg7xo625byk7x7uom

3D Shape Description and Matching Based on Properties of Real Functions [article]

Silvia Biasotti, Bianca Falcidieno, Patrizio Frosini, Daniela Giorgi, Claudia Landi, Simone Marini, Giuseppe Patané, Michela Spagnuolo
2007 Eurographics State of the Art Reports  
the mapping function, shape description, via the construction of a signature, and comparison, via the definition of a distance measure.  ...  The methods are discussed following an abstract conceptual framework that distinguishes among the three main components of these class of shape matching methods: shape analysis, via the application of  ...  Acknowledgements This work has been partially supported by the EC-IST FP6 Network of Excellence AIM@SHAPE. Models are courtesy of the AIM@SHAPE repository http://www.aimatshape.net.  ... 
doi:10.2312/egt.20071071 fatcat:kunadx2snzhwzncsxyla675kfy

Conceptualization of Object Compositions Using Persistent Homology [article]

Christian A. Mueller, Andreas Birk
2018 arXiv   pre-print
Inspired by Persistent Homology, hidden groups of shape commonalities are revealed from object segment decompositions.  ...  space in which segment decompositions are exposed in.  ...  Persistent Homology allows to track the birth and death of simplices in K vr of X during the filtration.  ... 
arXiv:1803.02140v3 fatcat:dz73pfzogjgbzpqsap2zdlytvu

Reviews: Topological Distances and Losses for Brain Networks [article]

Moo K. Chung, Alexander Smith, Gary Shiu
2021 arXiv   pre-print
In this review paper, we survey various topological distance and loss functions from topological data analysis (TDA) and persistent homology that can be used in brain network analysis more effectively.  ...  The Euclidean or matrix distances may fail to capture underlying subtle topological differences in brain networks. Further, Euclidean distances are sensitive to outliers.  ...  We would like to thank Andery Gritsenko of Northeastern University for the computation of the largest cycle in a graph.  ... 
arXiv:2102.08623v1 fatcat:vxddeghphfefbbt6k2zhdzidgu

Finding hidden order in spin models with persistent homology [article]

Bart Olsthoorn, Johan Hellsvik, Alexander V. Balatsky
2020 arXiv   pre-print
Persistent homology (PH) is a relatively new field in applied mathematics that studies the components and shapes of discrete data.  ...  Using dimensionality reduction to reduce the barcode space to color space leads to a visualization of the phase diagram.  ...  ACKNOWLEDGEMENTS The authors are grateful to M. Geilhufe, Jens H. Bar  ... 
arXiv:2009.05141v1 fatcat:x6ovbh7eebe3lbfs65ajx7brxu

Finding hidden order in spin models with persistent homology

Bart Olsthoorn, Johan Hellsvik, Alexander V. Balatsky
2020 Zenodo  
Persistent homology (PH) is a relatively new field in applied mathematics that studies the components and shapes of discrete data.  ...  Using dimensionality reduction to reduce the barcode space to color space leads to a visualization of the phase diagram.  ...  This could be improved by a faster distance metric for persistence diagrams, which is still an open problem in persistent homology.  ... 
doi:10.5281/zenodo.4282239 fatcat:z6bq2l7wfnfzthfdsia6ygdoza

An Introduction to Multiparameter Persistence [article]

Magnus Bakke Botnan, Michael Lesnick
2022 arXiv   pre-print
In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology.  ...  In this article, we introduce multiparameter persistence and survey some of this recent progress, with a focus on ideas likely to lead to practical applications in the near future.  ...  We also thank the authors of [170] for helpful feedback on our discussion of their work in Section 4.4 and for sharing Fig. 4 .3 with us.  ... 
arXiv:2203.14289v1 fatcat:ylrfxfdljrcidondwsiiqytea4

AffectiveTDA: Using Topological Data Analysis to Improve Analysis and Explainability in Affective Computing [article]

Hamza Elhamdadi and Shaun Canavan and Paul Rosen
2021 arXiv   pre-print
We present an approach utilizing Topological Data Analysis to study the structure of face poses used in affective computing, i.e., the process of recognizing human emotion.  ...  The approach uses a conditional comparison of different emotions, both respective and irrespective of time, with multiple topological distance metrics, dimension reduction techniques, and face subsections  ...  ACKNOWLEDGMENTS We thank the anonymous reviewers for their feedback. This project was supported in part by the National Science Foundation (IIS-1845204).  ... 
arXiv:2107.08573v2 fatcat:v4ap4aqjavgmlk6ly7mxnjgen4
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