{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T14:57:27Z","timestamp":1777129047162,"version":"3.51.4"},"reference-count":49,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T00:00:00Z","timestamp":1688601600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Appl. Math. Stat."],"abstract":"<jats:p>Persistent homology (PH) is a robust method to compute multi-dimensional geometric and topological features of a dataset. Because these features are often stable under certain perturbations of the underlying data, are often discriminating, and can be used for visualization of structure in high-dimensional data and in statistical and machine learning modeling, PH has attracted the interest of researchers across scientific disciplines and in many industry applications. However, computational costs may present challenges to effectively using PH in certain data contexts, and theoretical stability results may not hold in practice. In this paper, we define, implement, and investigate a simplicial complex construction for computing persistent homology of Euclidean point cloud data, which we call the Delaunay-Rips complex (DR). By only considering simplices that appear in the Delaunay triangulation of the point cloud and assigning the Vietoris-Rips weights to simplices, DR avoids potentially costly computations in the persistence calculations. We document and compare a Python implementation of DR with other simplicial complex constructions for generating persistence diagrams. By imposing sufficient conditions on point cloud data, we are able to theoretically justify the stability of the persistence diagrams produced using DR. When the Delaunay triangulation of the point cloud changes under perturbations of the points, we prove that DR-produced persistence diagrams exhibit instability. Since we cannot guarantee that real-world data will satisfy our stability conditions, we demonstrate the practical robustness of DR for persistent homology in comparison with other simplicial complexes in machine learning applications. We find in our experiments that using DR in an ML-TDA pipeline performs comparatively well as using other simplicial complex constructions.<\/jats:p>","DOI":"10.3389\/fams.2023.1179301","type":"journal-article","created":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T19:16:08Z","timestamp":1688670968000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":8,"title":["Stability and machine learning applications of persistent homology using the Delaunay-Rips complex"],"prefix":"10.3389","volume":"9","author":[{"given":"Amish","family":"Mishra","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francis C.","family":"Motta","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1965","published-online":{"date-parts":[[2023,7,6]]},"reference":[{"key":"B1","doi-asserted-by":"publisher","first-page":"20589","DOI":"10.1073\/pnas.0802806106","article-title":"Topological structures in the equities market network","volume":"105","author":"Leibon","year":"2008","journal-title":"Proc Nat Acad Sci"},{"key":"B2","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1007\/978-3-642-02498-6_32","article-title":"Persistence diagrams of cortical surface data","volume-title":"International Conference on Information Processing in Medical Imaging","author":"Chung","year":"2009"},{"key":"B3","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1016\/j.procs.2016.07.033","article-title":"Persistent homology for fast tumor segmentation in whole slide histology images","volume":"90","author":"Qaiser","year":"2016","journal-title":"Procedia Comput Sci"},{"key":"B4","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1007\/978-3-319-64185-0_11","article-title":"Topological Data Analysis for Image Tampering Detection","volume-title":"Digital Forensics and Watermarking","author":"Asaad","year":"2017"},{"key":"B5","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/j.physd.2014.05.009","article-title":"Quantifying force networks in particulate systems","volume":"283","author":"Kram\u00e1r","year":"2014","journal-title":"Physica D"},{"key":"B6","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1007\/BF02574053","article-title":"The union of balls and its dual shape","volume":"13","author":"Edelsbrunner","year":"1993","journal-title":"Discr Comput Geom"},{"key":"B7","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1145\/2261250.2261286","article-title":"Linear-size approximations to the Vietoris-Rips filtration","author":"Sheehy","year":"2012","journal-title":"Proceedings of the twenty-eighth annual symposium on Computational geometry"},{"key":"B8","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1007\/s00454-008-9094-6","article-title":"Reconstruction using witness complexes","volume":"40","author":"Guibas","year":"2007","journal-title":"Discr Comput Geom"},{"key":"B9","article-title":"Topological estimation using witness complexes","author":"de Silva","year":"2004","journal-title":"Symposium on Point Based Graphics"},{"key":"B10","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1007\/s41468-021-00071-5","article-title":"Ripser: efficient computation of Vietoris-Rips persistence barcodes","volume":"5","author":"Bauer","year":"2021","journal-title":"J Appl Comput Topol"},{"key":"B11","doi-asserted-by":"publisher","first-page":"925","DOI":"10.21105\/joss.00925","article-title":"Ripserpy: A lean persistent homology library for python","volume":"3","author":"Tralie","year":"2018","journal-title":"J Open Source Softw"},{"key":"B12","doi-asserted-by":"publisher","first-page":"2614","DOI":"10.21105\/joss.02614","article-title":"Ripserer jl: flexible and efficient persistent homology computation in Julia","volume":"5","author":"Cufar","year":"2020","journal-title":"J Open Source Software"},{"key":"B13","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1007\/s10711-008-9261-1","article-title":"A weak characterisation of the Delaunay triangulation","volume":"135","author":"Silva","year":"2008","journal-title":"Geometriae Dedicata"},{"key":"B14","doi-asserted-by":"publisher","first-page":"1393","DOI":"10.1111\/j.1467-8659.2009.01516.x","article-title":"Gromov-hausdorff stable signatures for shapes using persistence","volume":"28","author":"Chazal","year":"2009","journal-title":"Comput Graph Forum"},{"key":"B15","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/s00454-006-1276-5","article-title":"Stability of persistence diagrams","volume":"37","author":"Cohen-Steiner","year":"2005","journal-title":"Discr Comput Geom"},{"key":"B16","article-title":"Wasserstein stability for persistence diagrams","author":"Skraba","year":"2020","journal-title":"arXiv preprint arXiv:2006"},{"key":"B17","unstructured":"Cechmate\n            TralieC\n            SaulN\n          2021"},{"key":"B18","first-page":"1","article-title":"Persistence images: A stable vector representation of persistent homology","volume":"18","author":"Adams","year":"2017","journal-title":"J Mach Learn Res"},{"key":"B19","doi-asserted-by":"publisher","first-page":"637684","DOI":"10.3389\/fphys.2021.637684","article-title":"A persistent homology approach to heart rate variability analysis with an application to sleep-wake classification","volume":"12","author":"Chung","year":"2021","journal-title":"Front Physiol"},{"key":"B20","volume-title":"Computational Topology: An Introduction","author":"Edelsbrunner","year":"2010"},{"key":"B21","volume-title":"Algebraic Topology","author":"Hatcher","year":"2002"},{"key":"B22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1140\/epjds\/s13688-017-0109-5","article-title":"A roadmap for the computation of persistent homology","volume":"6","author":"Otter","year":"2017","journal-title":"EPJ Data Sci"},{"key":"B23","doi-asserted-by":"publisher","first-page":"3","DOI":"10.11575\/cdm.v3i2.61933","volume":"1","author":"Matouek","year":"2008","journal-title":"Contrib Discr Mathem"},{"key":"B24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.aam.2016.08.007","article-title":"Random cyclic dynamical systems","volume":"83","author":"Adamaszek","year":"2017","journal-title":"Adv Appl Math"},{"key":"B25","doi-asserted-by":"publisher","first-page":"3741","DOI":"10.1090\/tran\/6991","article-title":"The Morse theory of Tech and Delaunay complexes","volume":"369","author":"Bauer","year":"2016","journal-title":"Trans Am Mathem Soc"},{"key":"B26","unstructured":"MishraA,. Delaunay-Rips.\n          2022"},{"key":"B27","doi-asserted-by":"publisher","first-page":"76","DOI":"10.1016\/j.jsc.2016.03.008","article-title":"Phat persistent homology algorithms toolbox","volume":"78","author":"Bauer","year":"2017","journal-title":"J Symb Comput"},{"key":"B28","article-title":"Alpha complex","author":"Rouvreau","year":"2023","journal-title":"GUDHI User and Reference Manual 3rd ed GUDHI, Editorial Board"},{"key":"B29","article-title":"Stability of voronoi neighborship under perturbations of the sites","author":"Weller","year":"1997","journal-title":"CCCG"},{"key":"B30","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1142\/S0218195913600078","article-title":"The stability of Delaunay triangulations","volume":"23","author":"Boissonnat","year":"2013","journal-title":"Int J Comput Geom Appl"},{"key":"B31","unstructured":"2022"},{"key":"B32","unstructured":"SaulN\n            TralieC\n            MottaF\n            CatanzaroM\n            AngeloroG\n            SheagrenC.\n            Persim\n          2021"},{"key":"B33","first-page":"2825","article-title":"Scikit-learn: Machine Learning in Python","volume":"12","author":"Pedregosa","year":"2011","journal-title":"J Mach Learn Res"},{"key":"B34","doi-asserted-by":"publisher","first-page":"1135","DOI":"10.1145\/2939672.2939778","article-title":"\u201cWhy should i trust you?\u201d Explaining the predictions of any classifier.","author":"Ribeiro","year":"2016","journal-title":"Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining"},{"key":"B35","doi-asserted-by":"publisher","first-page":"757","DOI":"10.1038\/nmeth.4370","article-title":"Points of Significance: Classification and regression trees","volume":"14","author":"Krzywinski","year":"2017","journal-title":"Nat Methods"},{"key":"B36","unstructured":"MishraA,. ML-Del-Rips-sleep-wake-classification.\n          2022"},{"key":"B37","doi-asserted-by":"publisher","first-page":"085004","DOI":"10.1088\/1361-6579\/aad5a9","article-title":"Wu Ht Sleep-wake classification via quantifying heart rate variability by convolutional neural network","volume":"39","author":"Malik","year":"2018","journal-title":"Physiol Measur"},{"key":"B38","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1007\/BFb0091924","article-title":"Detecting strange attractors in turbulence","volume-title":"Dynamical Systems and Turbulence, Warwick 1980","author":"Takens","year":"1981"},{"key":"B39","doi-asserted-by":"publisher","first-page":"4150","DOI":"10.1109\/ICIP.2016.7533141","article-title":"Persistent homology of attractors for action recognition","author":"Venkataraman","year":"2016","journal-title":"2016 IEEE International Conference on Image Processing (ICIP)"},{"key":"B40","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1054\/jelc.2001.28876","article-title":"Automatic detection of artifacts in heart period data","volume":"34","author":"Xu","year":"2001","journal-title":"J Electrocardiol"},{"key":"B41","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1016\/j.patcog.2014.06.023","article-title":"An entropy-based persistence barcode","volume":"48","author":"Chintakunta","year":"2015","journal-title":"Patt Recognit"},{"key":"B42","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1038\/s41592-019-0686-2","article-title":"SciPy 10: fundamental algorithms for scientific computing in python","volume":"17","author":"Virtanen","year":"2020","journal-title":"Nature Methods"},{"key":"B43","unstructured":"JavaPlex: A research software package for persistent (co)homology.129136\n            TauszA\n            Vejdemo-JohanssonM\n            AdamsH\n          \n            HongH\n            YapC\n          10.1007\/978-3-662-44199-2_23Proceedings of ICMS 2014 Lecture Notes in Computer Science2014"},{"key":"B44","unstructured":"NandaV.\n          Perseus, the Persistent Homology Software.2013"},{"key":"B45","unstructured":"MorozovD.\n          Dionysus.2023"},{"key":"B46","unstructured":"ReininghausJ.\n          DIPHA (A Distributed Persistent Homology Algorithm).2017"},{"key":"B47","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/978-3-662-44199-2_28","article-title":"The gudhi library: Simplicial complexes and persistent homology","volume-title":"Mathematical Software ICMS 2014: 4th International Congress, Seoul, South Korea","author":"Maria","year":"2014"},{"key":"B48","article-title":"GPU-accelerated computation of Vietoris-Rips persistence barcodes","author":"Simon Zhang","year":"2020","journal-title":"arXiv preprint arXiv:2003"},{"key":"B49","article-title":"Stability and machine learning applications of persistent homology using the Delaunay-Rips complex","author":"Mishra","year":"2023","journal-title":"arXiv preprint arXiv:2303"}],"container-title":["Frontiers in Applied Mathematics and Statistics"],"original-title":[],"link":[{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fams.2023.1179301\/full","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T19:16:25Z","timestamp":1688670985000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fams.2023.1179301\/full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,6]]},"references-count":49,"alternative-id":["10.3389\/fams.2023.1179301"],"URL":"https:\/\/doi.org\/10.3389\/fams.2023.1179301","relation":{},"ISSN":["2297-4687"],"issn-type":[{"value":"2297-4687","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,6]]},"article-number":"1179301"}}