{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T22:59:58Z","timestamp":1740178798143,"version":"3.37.3"},"reference-count":55,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,6,28]],"date-time":"2024-06-28T00:00:00Z","timestamp":1719532800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,6,28]],"date-time":"2024-06-28T00:00:00Z","timestamp":1719532800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["GRFP-1650116"],"award-info":[{"award-number":["GRFP-1650116"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Appl. and Comput. Topology"],"published-print":{"date-parts":[[2024,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Commutative diagrams of vector spaces and linear maps over <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {Z}^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>Z<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are objects of interest in topological data analysis (TDA) where this type of diagrams are called 2-parameter persistence modules. Given that quiver representation theory tells us that such diagrams are of wild type, studying informative invariants of a 2-parameter persistence module <jats:italic>M<\/jats:italic> is of central importance in TDA. One of such invariants is the generalized rank invariant, recently introduced by Kim and M\u00e9moli. Via the M\u00f6bius inversion of the generalized rank invariant of <jats:italic>M<\/jats:italic>, we obtain a collection of connected subsets <jats:inline-formula><jats:alternatives><jats:tex-math>$$I\\subset \\mathbb {Z}^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>I<\/mml:mi>\n                    <mml:mo>\u2282<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mi>Z<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with signed multiplicities. This collection generalizes the well known notion of <jats:italic>persistence barcode<\/jats:italic> of a persistence module over <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {R}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>R<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> from TDA. In this paper we show that the bigraded Betti numbers of <jats:italic>M<\/jats:italic>, a classical algebraic invariant of <jats:italic>M<\/jats:italic>, are obtained by counting the corner points of these subsets <jats:italic>I<\/jats:italic>s. Along the way, we verify that an invariant of 2-parameter persistence modules called the interval decomposable approximation (introduced by Asashiba et al.) also encodes the bigraded Betti numbers in a similar fashion. We also show that the aforementioned results are optimal in the sense that they cannot be extended to <jats:italic>d<\/jats:italic>-parameter persistence modules for <jats:inline-formula><jats:alternatives><jats:tex-math>$$d \\ge 3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s41468-024-00180-x","type":"journal-article","created":{"date-parts":[[2024,6,28]],"date-time":"2024-06-28T04:03:28Z","timestamp":1719547408000},"page":"727-760","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bigraded Betti numbers and generalized persistence diagrams"],"prefix":"10.1007","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8081-5872","authenticated-orcid":false,"given":"Woojin","family":"Kim","sequence":"first","affiliation":[]},{"given":"Samantha","family":"Moore","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,6,28]]},"reference":[{"key":"180_CR1","unstructured":"Asashiba, H., Escolar, E.G., Nakashima, K., Yoshiwaki, M.: On approximation of $$2$$ d persistence modules by interval-decomposables. arXiv preprint arXiv:1911.01637 (2019)"},{"key":"180_CR2","doi-asserted-by":"publisher","DOI":"10.1016\/j.comgeo.2022.101879","volume":"105\u2013106","author":"H Asashiba","year":"2022","unstructured":"Asashiba, H., Buchet, M., Escolar, E.G., Nakashima, K., Yoshiwaki, M.: On interval decomposability of 2d persistence modules. Comput. Geom. 105\u2013106, 101879 (2022). https:\/\/doi.org\/10.1016\/j.comgeo.2022.101879","journal-title":"Comput. Geom."},{"key":"180_CR3","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1017\/S002776300002290X","volume":"1","author":"G Azumaya","year":"1950","unstructured":"Azumaya, G., et al.: Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt\u2019s theorem. Nagoya Math. J. 1, 117\u2013124 (1950)","journal-title":"Nagoya Math. J."},{"key":"180_CR4","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2020.107171","volume":"369","author":"U Bauer","year":"2020","unstructured":"Bauer, U., Botnan, M.B., Oppermann, S., Steen, J.: Cotorsion torsion triples and the representation theory of filtered hierarchical clustering. Adv. Math. 369, 107171 (2020)","journal-title":"Adv. Math."},{"issue":"8","key":"180_CR5","first-page":"789","volume":"82","author":"EA Bender","year":"1975","unstructured":"Bender, E.A., Goldman, J.R.: On the applications of M\u00f6bius inversion in combinatorial analysis. Am. Math. Mon. 82(8), 789\u2013803 (1975)","journal-title":"Am. Math. Mon."},{"key":"180_CR6","unstructured":"Blanchette, B., Br\u00fcstle, T., Hanson, E.J.: Homological approximations in persistence theory. arXiv preprint arXiv:2112.07632 (2021)"},{"key":"180_CR7","doi-asserted-by":"publisher","unstructured":"Botnan, M.B., Lebovici, V., Oudot, S.: On rectangle-decomposable 2-parameter persistence modules. In: 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), vol. 164, pp. 22\u201312216. (2020). Schloss Dagstuhl\u2013Leibniz-Zentrum f\u00fcr Informatik, Dagstuhl, Germany. https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2020.22. https:\/\/drops.dagstuhl.de\/opus\/volltexte\/2020\/12180","DOI":"10.4230\/LIPIcs.SoCG.2020.22"},{"key":"180_CR8","unstructured":"Botnan, M.B., Oppermann, S., Oudot, S.: Signed barcodes for multi-parameter persistence via rank decompositions and rank-exact resolutions. arXiv preprint arXiv:2107.06800 (2021)"},{"issue":"11","key":"180_CR9","doi-asserted-by":"publisher","first-page":"4581","DOI":"10.1090\/proc\/14790","volume":"148","author":"M Botnan","year":"2020","unstructured":"Botnan, M., Crawley-Boevey, W.: Decomposition of persistence modules. Proceed. Am. Math. Soc. 148(11), 4581\u20134596 (2020)","journal-title":"Proceed. Am. Math. Soc."},{"issue":"6","key":"180_CR10","doi-asserted-by":"publisher","first-page":"3133","DOI":"10.2140\/agt.2018.18.3133","volume":"18","author":"M Botnan","year":"2018","unstructured":"Botnan, M., Lesnick, M.: Algebraic stability of zigzag persistence modules. Algebr. Geom. Topol. 18(6), 3133\u20133204 (2018)","journal-title":"Algebr. Geom. Topol."},{"issue":"3","key":"180_CR11","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1137\/20M1353605","volume":"5","author":"C Cai","year":"2021","unstructured":"Cai, C., Kim, W., M\u00e9moli, F., Wang, Y.: Elder-rule-staircodes for augmented metric spaces. SIAM J. Appl. Algebra Geomet. 5(3), 417\u2013454 (2021)","journal-title":"SIAM J. Appl. Algebra Geomet."},{"key":"180_CR12","doi-asserted-by":"crossref","unstructured":"Carlsson, G., De\u00a0Silva, V., Morozov, D.: Zigzag persistent homology and real-valued functions. In: Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, pp. 247\u2013256 (2009)","DOI":"10.1145\/1542362.1542408"},{"key":"180_CR13","doi-asserted-by":"crossref","unstructured":"Carlsson, G., M\u00e9moli, F.: Multiparameter hierarchical clustering methods. In: Classification as a Tool for Research, pp. 63\u201370. Springer, Heidelberg (2010)","DOI":"10.1007\/978-3-642-10745-0_6"},{"issue":"2","key":"180_CR14","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1090\/S0273-0979-09-01249-X","volume":"46","author":"G Carlsson","year":"2009","unstructured":"Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255\u2013308 (2009)","journal-title":"Bull. Am. Math. Soc."},{"issue":"4","key":"180_CR15","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1007\/s10208-010-9066-0","volume":"10","author":"G Carlsson","year":"2010","unstructured":"Carlsson, G., De Silva, V.: Zigzag persistence. Found. Comput. Math. 10(4), 367\u2013405 (2010)","journal-title":"Found. Comput. Math."},{"issue":"1","key":"180_CR16","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1007\/s00454-009-9176-0","volume":"42","author":"G Carlsson","year":"2009","unstructured":"Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. Discrete Comput. Geomet. 42(1), 71\u201393 (2009)","journal-title":"Discrete Comput. Geomet."},{"issue":"02","key":"180_CR17","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1142\/S0218654305000761","volume":"11","author":"G Carlsson","year":"2005","unstructured":"Carlsson, G., Zomorodian, A., Collins, A., Guibas, L.J.: Persistence barcodes for shapes. Int. J. Shape Model. 11(02), 149\u2013187 (2005)","journal-title":"Int. J. Shape Model."},{"issue":"5","key":"180_CR18","doi-asserted-by":"publisher","first-page":"1055","DOI":"10.1016\/j.jpaa.2016.09.001","volume":"221","author":"W Chach\u00f3lski","year":"2017","unstructured":"Chach\u00f3lski, W., Scolamiero, M., Vaccarino, F.: Combinatorial presentation of multidimensional persistent homology. J. Pure Appl. Algebra 221(5), 1055\u20131075 (2017)","journal-title":"J. Pure Appl. Algebra"},{"key":"180_CR19","doi-asserted-by":"crossref","unstructured":"Chambers, E., Letscher, D.: Persistent homology over directed acyclic graphs. In: Research in Computational Topology, pp. 11\u201332. Springer, Switzerland (2018)","DOI":"10.1007\/978-3-319-89593-2_2"},{"issue":"2","key":"180_CR20","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1007\/s00454-019-00165-z","volume":"63","author":"J Cochoy","year":"2020","unstructured":"Cochoy, J., Oudot, S.: Decomposition of exact pfd persistence bimodules. Discrete Comput. Geomet. 63(2), 255\u2013293 (2020)","journal-title":"Discrete Comput. Geomet."},{"issue":"1","key":"180_CR21","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/s00454-006-1276-5","volume":"37","author":"D Cohen-Steiner","year":"2007","unstructured":"Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geomet. 37(1), 103\u2013120 (2007)","journal-title":"Discrete Comput. Geomet."},{"issue":"05","key":"180_CR22","doi-asserted-by":"publisher","first-page":"1550066","DOI":"10.1142\/S0219498815500668","volume":"14","author":"W Crawley-Boevey","year":"2015","unstructured":"Crawley-Boevey, W.: Decomposition of pointwise finite-dimensional persistence modules. J. Algebra Appl. 14(05), 1550066 (2015)","journal-title":"J. Algebra Appl."},{"issue":"2","key":"180_CR23","first-page":"200","volume":"52","author":"H Derksen","year":"2005","unstructured":"Derksen, H., Weyman, J.: Quiver representations. Notices AMS 52(2), 200\u2013206 (2005)","journal-title":"Notices AMS"},{"key":"180_CR24","doi-asserted-by":"publisher","unstructured":"Dey, T.K., Kim, W., M\u00e9moli, F.: Computing generalized rank invariant for 2-parameter persistence modules via zigzag persistence and its applications. In: 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), vol. 224, pp. 34\u201313417. Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik, Dagstuhl, Germany (2022). https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2022.34. https:\/\/drops.dagstuhl.de\/opus\/volltexte\/2022\/16042","DOI":"10.4230\/LIPIcs.SoCG.2022.34"},{"key":"180_CR25","doi-asserted-by":"publisher","unstructured":"Dey, T.K., Xin, C.: Computing Bottleneck Distance for 2-D Interval Decomposable Modules. In: 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), vol. 99, pp. 32\u201313215. Schloss Dagstuhl\u2013Leibniz-Zentrum fuer Informatik, (2018). https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2018.32. http:\/\/drops.dagstuhl.de\/opus\/volltexte\/2018\/8745","DOI":"10.4230\/LIPIcs.SoCG.2018.32"},{"key":"180_CR26","doi-asserted-by":"crossref","unstructured":"Dey, T.K., Xin, C.: Generalized persistence algorithm for decomposing multiparameter persistence modules. Journal of Applied and Computational Topology, 1\u201352 (2022)","DOI":"10.1007\/s41468-022-00087-5"},{"key":"180_CR27","unstructured":"Dey, T.K., Xin, C.: Rectangular approximation and stability of $$2 $$-parameter persistence modules. arXiv preprint. (2021). arXiv:2108.07429"},{"key":"180_CR28","doi-asserted-by":"crossref","unstructured":"Edelsbrunner, H., Harer, J.: Computational Topology: an Introduction. American Mathematical Soc., (2010)","DOI":"10.1090\/mbk\/069"},{"key":"180_CR29","unstructured":"Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Proceedings 41st Annual Symposium on Foundations of Computer Science, pp. 454\u2013463 (2000). IEEE"},{"key":"180_CR30","volume-title":"The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra","author":"D Eisenbud","year":"2002","unstructured":"Eisenbud, D.: The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra. Springer, New York (2002)"},{"issue":"1","key":"180_CR31","doi-asserted-by":"publisher","first-page":"100","DOI":"10.1007\/s00454-015-9746-2","volume":"55","author":"EG Escolar","year":"2016","unstructured":"Escolar, E.G., Hiraoka, Y.: Persistence modules on commutative ladders of finite type. Discrete Comput. Geomet. 55(1), 100\u2013157 (2016)","journal-title":"Discrete Comput. Geomet."},{"key":"180_CR32","doi-asserted-by":"crossref","unstructured":"Gabriel, P.: Unzerlegbare darstellungen i. Manuscripta Mathematica, 71\u2013103 (1972)","DOI":"10.1007\/BF01298413"},{"issue":"3","key":"180_CR33","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1137\/18M1224350","volume":"3","author":"HA Harrington","year":"2019","unstructured":"Harrington, H.A., Otter, N., Schenck, H., Tillmann, U.: Stratifying multiparameter persistent homology. SIAM J. Appl. Algebra Geomet. 3(3), 439\u2013471 (2019)","journal-title":"SIAM J. Appl. Algebra Geomet."},{"key":"180_CR34","doi-asserted-by":"crossref","unstructured":"Hilbert, D.: \u00dcuber die theorie von algebraischen formen. Mathematische Annalen, 473\u2013534 (1890)","DOI":"10.1007\/BF01208503"},{"issue":"26","key":"180_CR35","doi-asserted-by":"publisher","first-page":"7035","DOI":"10.1073\/pnas.1520877113","volume":"113","author":"Y Hiraoka","year":"2016","unstructured":"Hiraoka, Y., Nakamura, T., Hirata, A., Escolar, E.G., Matsue, K., Nishiura, Y.: Hierarchical structures of amorphous solids characterized by persistent homology. Proc. Natl. Acad. Sci. 113(26), 7035\u20137040 (2016)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"180_CR36","doi-asserted-by":"publisher","unstructured":"Keller, B., Lesnick, M., Willke, T.L.: Persistent homology for virtual screening. ChemRxiv.6969260.v3 (2018). https:\/\/doi.org\/10.26434\/chemrxiv.6969260.v3","DOI":"10.26434\/chemrxiv.6969260.v3"},{"key":"180_CR37","doi-asserted-by":"crossref","unstructured":"Kerber, M., Rolle, A.: Fast minimal presentations of bi-graded persistence modules. In: 2021 Proceedings of the Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 207\u2013220. SIAM (2021)","DOI":"10.1137\/1.9781611976472.16"},{"issue":"4","key":"180_CR38","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1007\/s41468-021-00075-1","volume":"5","author":"W Kim","year":"2021","unstructured":"Kim, W., M\u00e9moli, F.: Generalized persistence diagrams for persistence modules over posets. J. Appl. Computat. Topol. 5(4), 533\u2013581 (2021)","journal-title":"J. Appl. Computat. Topol."},{"issue":"1","key":"180_CR39","doi-asserted-by":"publisher","first-page":"259","DOI":"10.4310\/HHA.2008.v10.n1.a11","volume":"10","author":"KP Knudson","year":"2008","unstructured":"Knudson, K.P.: A refinement of multi-dimensional persistence. Homology Homotopy Appl. 10(1), 259\u2013281 (2008)","journal-title":"Homology Homotopy Appl."},{"key":"180_CR40","unstructured":"Landi, C., Frosini, P.: New pseudodistances for the size function space. In: Vision Geometry VI, vol. 3168, pp. 52\u201360 (1997). International Society for Optics and Photonics"},{"key":"180_CR41","unstructured":"Lesnick, M., Wright, M.: Interactive visualization of 2-d persistence modules. arXiv preprint arXiv:1512.00180 (2015)"},{"issue":"3","key":"180_CR42","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1007\/s10208-015-9255-y","volume":"15","author":"M Lesnick","year":"2015","unstructured":"Lesnick, M.: The theory of the interleaving distance on multidimensional persistence modules. Found. Comput. Math. 15(3), 613\u2013650 (2015)","journal-title":"Found. Comput. Math."},{"issue":"2","key":"180_CR43","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1137\/20M1388425","volume":"6","author":"M Lesnick","year":"2022","unstructured":"Lesnick, M., Wright, M.: Computing minimal presentations and bigraded betti numbers of 2-parameter persistent homology. SIAM J. Appl. Algebra Geomet. 6(2), 267\u2013298 (2022)","journal-title":"SIAM J. Appl. Algebra Geomet."},{"key":"180_CR44","volume-title":"Categories for the Working Mathematician","author":"S Mac Lane","year":"2013","unstructured":"Mac Lane, S.: Categories for the Working Mathematician, vol. 5. Springer, New York (2013)"},{"issue":"7","key":"180_CR45","doi-asserted-by":"publisher","first-page":"3149","DOI":"10.1090\/proc\/14929","volume":"148","author":"A McCleary","year":"2020","unstructured":"McCleary, A., Patel, A.: Bottleneck stability for generalized persistence diagrams. Proceed. Am. Math. Soc. 148(7), 3149\u20133161 (2020)","journal-title":"Proceed. Am. Math. Soc."},{"issue":"2","key":"180_CR46","doi-asserted-by":"publisher","first-page":"134","DOI":"10.1137\/20M1373700","volume":"6","author":"A McCleary","year":"2022","unstructured":"McCleary, A., Patel, A.: Edit distance and persistence diagrams over lattices. SIAM J. Appl. Algebra Geomet. 6(2), 134\u2013155 (2022)","journal-title":"SIAM J. Appl. Algebra Geomet."},{"key":"180_CR47","unstructured":"Miller, E.: Homological algebra of modules over posets. arXiv preprint arXiv:2008.00063 (2020)"},{"key":"180_CR48","volume-title":"Combinatorial Commutative Algebra","author":"E Miller","year":"2005","unstructured":"Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra, vol. 227. Springer, New York (2005)"},{"key":"180_CR49","unstructured":"Moore, S.: A combinatorial formula for the bigraded betti numbers. arXiv preprint arXiv:2004.02239 (2020)"},{"key":"180_CR50","unstructured":"Moore, S.: On the structure of multiparameter persistence modules. PhD thesis, University of North Carolina at Chapel Hill (2022)"},{"issue":"3","key":"180_CR51","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1007\/s41468-018-0012-6","volume":"1","author":"A Patel","year":"2018","unstructured":"Patel, A.: Generalized persistence diagrams. J. Appl. Computat. Topol. 1(3), 397\u2013419 (2018)","journal-title":"J. Appl. Computat. Topol."},{"key":"180_CR52","doi-asserted-by":"crossref","unstructured":"Rota, G.C.: On the foundations of combinatorial theory i. theory of m\u00f6bius functions. Zeitschrift f\u00fcr Wahrscheinlichkeitstheorie und verwandte Gebiete 2(4), 340\u2013368 (1964)","DOI":"10.1007\/BF00531932"},{"issue":"6","key":"180_CR53","doi-asserted-by":"publisher","first-page":"1367","DOI":"10.1007\/s10208-016-9323-y","volume":"17","author":"M Scolamiero","year":"2017","unstructured":"Scolamiero, M., Chach\u00f3lski, W., Lundman, A., Ramanujam, R., \u00d6berg, S.: Multidimensional persistence and noise. Found. Comput. Math. 17(6), 1367\u20131406 (2017)","journal-title":"Found. Comput. Math."},{"key":"180_CR54","unstructured":"Thomas, A.L.: Invariants and metrics for multiparameter persistent homology. PhD thesis, Duke University (2019)"},{"key":"180_CR55","first-page":"61","volume":"21","author":"O Vipond","year":"2020","unstructured":"Vipond, O.: Multiparameter persistence landscapes. J. Mach. Learn. Res. 21, 61\u20131 (2020)","journal-title":"J. Mach. Learn. Res."}],"container-title":["Journal of Applied and Computational Topology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41468-024-00180-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s41468-024-00180-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41468-024-00180-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,20]],"date-time":"2024-09-20T15:10:00Z","timestamp":1726845000000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s41468-024-00180-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,28]]},"references-count":55,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,9]]}},"alternative-id":["180"],"URL":"https:\/\/doi.org\/10.1007\/s41468-024-00180-x","relation":{},"ISSN":["2367-1726","2367-1734"],"issn-type":[{"type":"print","value":"2367-1726"},{"type":"electronic","value":"2367-1734"}],"subject":[],"published":{"date-parts":[[2024,6,28]]},"assertion":[{"value":"1 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 May 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 May 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 June 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no Conflict of interest to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}