{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,21]],"date-time":"2026-03-21T19:42:51Z","timestamp":1774122171821,"version":"3.50.1"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,5,11]],"date-time":"2021-05-11T00:00:00Z","timestamp":1620691200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,5,11]],"date-time":"2021-05-11T00:00:00Z","timestamp":1620691200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"NTNU Norwegian University of Science and Technology"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2021,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The algebraic stability theorem for persistence modules is a central result in the theory of stability for persistent homology. We introduce a new proof technique which we use to prove a stability theorem for <jats:italic>n<\/jats:italic>-dimensional rectangle decomposable persistence modules up to a constant <jats:inline-formula><jats:alternatives><jats:tex-math>$$2n-1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that generalizes the algebraic stability theorem, and give an example showing that the bound cannot be improved for <jats:inline-formula><jats:alternatives><jats:tex-math>$$n=2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We then apply the technique to prove stability for block decomposable modules, from which novel results for zigzag modules and Reeb graphs follow. These results are improvements on weaker bounds in previous work, and the bounds we obtain are optimal.<\/jats:p>","DOI":"10.1007\/s00454-021-00298-0","type":"journal-article","created":{"date-parts":[[2021,5,11]],"date-time":"2021-05-11T13:03:15Z","timestamp":1620738195000},"page":"92-121","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["On the Stability of Interval Decomposable Persistence Modules"],"prefix":"10.1007","volume":"66","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9778-0354","authenticated-orcid":false,"given":"H\u00e5vard","family":"Bakke Bjerkevik","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,5,11]]},"reference":[{"key":"298_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-00856-6","volume-title":"Proofs from The Book","author":"M Aigner","year":"2010","unstructured":"Aigner, M., Ziegler, G.M.: Proofs from The Book. Springer, Berlin (2010)"},{"key":"298_CR2","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1017\/S002776300002290X","volume":"1","author":"G Azumaya","year":"1950","unstructured":"Azumaya, G.: Corrections and supplementaries to my paper concerning Krull\u2013Remak\u2013Schmidt\u2019s theorem. Nagoya Math. J. 1, 117\u2013124 (1950)","journal-title":"Nagoya Math. J."},{"issue":"2","key":"298_CR3","first-page":"162","volume":"6","author":"U Bauer","year":"2015","unstructured":"Bauer, U., Lesnick, M.: Induced matchings and the algebraic stability of persistence barcodes. J. Comput. Geom. 6(2), 162\u2013191 (2015)","journal-title":"J. Comput. Geom."},{"key":"298_CR4","unstructured":"Bauer, U., Munch, E., Wang, Y.: Strong equivalence of the interleaving and functional distortion metrics for Reeb graphs. In: 31st International Symposium on Computational Geometry. Leibniz Int. Proc. Inform., vol.\u00a034, pp. 461\u2013475. Leibniz-Zent. Inform., Wadern (2015)"},{"issue":"1","key":"298_CR5","doi-asserted-by":"publisher","first-page":"51","DOI":"10.4310\/HHA.2013.v15.n1.a3","volume":"15","author":"P Bendich","year":"2013","unstructured":"Bendich, P., Edelsbrunner, H., Morozov, D., Patel, A.: Homology and robustness of level and interlevel sets. Homol. Homotopy Appl. 15(1), 51\u201372 (2013)","journal-title":"Homol. Homotopy Appl."},{"key":"298_CR6","unstructured":"Bjerkevik, H.B., Botnan, M.B.: Computational complexity of the interleaving distance. In: Proceedings 34th International Symposium on Computational Geometry. Leibniz Int. Proc. Inform., vol. 99, # 13. Leibniz-Zent. Inform., Wadern (2018)"},{"issue":"5","key":"298_CR7","doi-asserted-by":"publisher","first-page":"1237","DOI":"10.1007\/s10208-019-09442-y","volume":"20","author":"HB Bjerkevik","year":"2020","unstructured":"Bjerkevik, H.B., Botnan, M.B., Kerber, M.: Computing the interleaving distance is NP-hard. Found. Comput. Math. 20(5), 1237\u20131271 (2020)","journal-title":"Found. Comput. Math."},{"key":"298_CR8","unstructured":"Botnan, M.B.: Applications and Generalizations of the Algebraic Stability Theorem. PhD thesis, Norwegian University of Science and Technology (2015)"},{"issue":"8","key":"298_CR9","doi-asserted-by":"publisher","first-page":"3571","DOI":"10.1090\/proc\/13465","volume":"145","author":"MB Botnan","year":"2017","unstructured":"Botnan, M.B.: Interval decomposition of infinite zigzag persistence modules. Proc. Am. Math. Soc. 145(8), 3571\u20133577 (2017)","journal-title":"Proc. Am. Math. Soc."},{"issue":"11","key":"298_CR10","doi-asserted-by":"publisher","first-page":"4581","DOI":"10.1090\/proc\/14790","volume":"148","author":"MB Botnan","year":"2020","unstructured":"Botnan, M.B., Crawley-Boevey, W.: Decomposition of persistence modules. Proc. Am. Math. Soc. 148(11), 4581\u20134596 (2020)","journal-title":"Proc. Am. Math. Soc."},{"issue":"6","key":"298_CR11","doi-asserted-by":"publisher","first-page":"3133","DOI":"10.2140\/agt.2018.18.3133","volume":"18","author":"MB Botnan","year":"2018","unstructured":"Botnan, M.B., Lesnick, M.: Algebraic stability of zigzag persistence modules. Algebr. Geom. Topol. 18(6), 3133\u20133204 (2018)","journal-title":"Algebr. Geom. Topol."},{"key":"298_CR12","series-title":"Data Analysis and Knowledge Organization","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/978-3-642-10745-0_6","volume-title":"Classification as a Tool for Research (Dresden 2009). Studies in Classification","author":"G Carlsson","year":"2010","unstructured":"Carlsson, G., M\u00e9moli, F.: Multiparameter hierarchical clustering methods. Classification as a Tool for Research (Dresden 2009). Studies in Classification. Data Analysis and Knowledge Organization, pp. 63\u201370. Springer, Berlin (2010)"},{"key":"298_CR13","doi-asserted-by":"crossref","unstructured":"Carlsson, G., de Silva, V., Morozov, D.: Zigzag persistent homology and real-valued functions. In: 25th Annual Symposium on Computational Geometry (Aarhus 2009), pp. 247\u2013256. ACM, New York (2009)","DOI":"10.1145\/1542362.1542408"},{"issue":"1","key":"298_CR14","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. Discret. Comput. Geom. 42(1), 71\u201393 (2009)","journal-title":"Discret. Comput. Geom."},{"key":"298_CR15","doi-asserted-by":"crossref","unstructured":"Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.Y.: Proximity of persistence modules and their diagrams. In: 25th Annual Symposium on Computational Geometry (Aarhus 2009), pp. 237\u2013246. ACM, New York (2009)","DOI":"10.1145\/1542362.1542407"},{"key":"298_CR16","series-title":"SpringerBriefs in Mathematics","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-42545-0","volume-title":"The Structure and Stability of Persistence Modules","author":"F Chazal","year":"2016","unstructured":"Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The Structure and Stability of Persistence Modules. SpringerBriefs in Mathematics. Springer, Cham (2016)"},{"issue":"2","key":"298_CR17","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. Discret. Comput. Geom. 63(2), 255\u2013293 (2020)","journal-title":"Discret. Comput. Geom."},{"issue":"1","key":"298_CR18","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. Discret. Comput. Geom. 37(1), 103\u2013120 (2007)","journal-title":"Discret. Comput. Geom."},{"key":"298_CR19","doi-asserted-by":"crossref","unstructured":"Crawley-Boevey, W.: Decomposition of pointwise finite-dimensional persistence modules. J. Algebra Appl. 14(5), # 1550066 (2015)","DOI":"10.1142\/S0219498815500668"},{"key":"298_CR20","unstructured":"Dey, T.K., Xin, C.: Computing bottleneck distance for 2-D interval decomposable modules. In: 34th International Symposium on Computational Geometry. Leibniz Int. Proc. Inform., vol. 99, #\u00a032. Leibniz-Zent. Inform., Wadern (2018)"},{"key":"298_CR21","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1112\/jlms\/s1-10.37.26","volume":"10","author":"P Hall","year":"1935","unstructured":"Hall, P.: On representatives of subsets. J. Lond. Math. Soc. 10, 26\u201330 (1935)","journal-title":"J. Lond. Math. Soc."},{"key":"298_CR22","doi-asserted-by":"crossref","unstructured":"Hilaga, M., Shinagawa, Y., Kohmura, T., Kunii, T.L.: Topology matching for fully automatic similarity estimation of 3D shapes. In: 28th Annual Conference on Computer Graphics and Interactive Techniques (Los Angeles 2001), pp. 203\u2013212. ACM, New York (2001)","DOI":"10.1145\/383259.383282"},{"key":"298_CR23","unstructured":"Kim, W., M\u00e9moli, F.: Stable signatures for dynamic metric spaces via zigzag persistent homology (2017).  arXiv:1712.04064"},{"issue":"5","key":"298_CR24","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/j.gmod.2011.03.002","volume":"73","author":"M Natali","year":"2011","unstructured":"Natali, M., Biasotti, S., Patan\u00e8, G., Falcidieno, B.: Graph-based representations of point clouds. Graph. Models 73(5), 151\u2013164 (2011)","journal-title":"Graph. Models"},{"issue":"17","key":"298_CR25","doi-asserted-by":"publisher","first-page":"7265","DOI":"10.1073\/pnas.1102826108","volume":"108","author":"M Nicolau","year":"2011","unstructured":"Nicolau, M., Levine, A.J., Carlsson, G.: Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proc. Natl. Acad. Sci. USA 108(17), 7265\u20137270 (2011)","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"298_CR26","first-page":"847","volume":"222","author":"G Reeb","year":"1946","unstructured":"Reeb, G.: Sur les points singuliers d\u2019une forme de Pfaff compl\u00e8tement int\u00e9grable ou d\u2019une fonction num\u00e9rique. C. R. Acad. Sci. Paris 222, 847\u2013849 (1946)","journal-title":"C. R. Acad. Sci. Paris"},{"issue":"5","key":"298_CR27","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1109\/38.90568","volume":"11","author":"Y Shinagawa","year":"1991","unstructured":"Shinagawa, Y., Kunii, T.L., Kergosien, Y.L.: Surface coding based on Morse theory. IEEE Comput. Graph. Appl. 11(5), 66\u201378 (1991)","journal-title":"IEEE Comput. Graph. Appl."},{"issue":"4","key":"298_CR28","doi-asserted-by":"publisher","first-page":"854","DOI":"10.1007\/s00454-016-9763-9","volume":"55","author":"V de Silva","year":"2016","unstructured":"de Silva, V., Munch, E., Patel, A.: Categorified Reeb graphs. Discret. Comput. Geom. 55(4), 854\u2013906 (2016)","journal-title":"Discret. Comput. Geom."},{"key":"298_CR29","unstructured":"Singh, G., M\u00e9moli, F., Carlsson, G.: Topological methods for the analysis of high dimensional data sets and 3D object recognition. In: Eurographics Symposium on Point-Based Graphics (Prague 2007), pp. 91\u2013100. Eurographics Association (2007)"},{"issue":"2","key":"298_CR30","doi-asserted-by":"publisher","first-page":"190","DOI":"10.1145\/990002.990007","volume":"23","author":"Z Wood","year":"2004","unstructured":"Wood, Z., Hoppe, H., Desbrun, M., Schr\u00f6der, P.: Removing excess topology from isosurfaces. ACM Trans. Graph. 23(2), 190\u2013208 (2004)","journal-title":"ACM Trans. Graph."},{"key":"298_CR31","doi-asserted-by":"crossref","unstructured":"Yao, Y., Sun, J., Huang, X., Bowman, G.R., Singh, G., Lesnick, M., Guibas, L.J., Pande, V.S., Carlsson, G.: Topological methods for exploring low-density states in biomolecular folding pathways. J. Chem. Phys. 130(14), #\u00a0144115 (2009)","DOI":"10.1063\/1.3103496"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-021-00298-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-021-00298-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-021-00298-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,9]],"date-time":"2021-06-09T15:09:50Z","timestamp":1623251390000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-021-00298-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,11]]},"references-count":31,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,7]]}},"alternative-id":["298"],"URL":"https:\/\/doi.org\/10.1007\/s00454-021-00298-0","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,11]]},"assertion":[{"value":"7 December 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 January 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 March 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 May 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}