{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T12:39:15Z","timestamp":1770467955204,"version":"3.49.0"},"reference-count":91,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,4,22]],"date-time":"2024-04-22T00:00:00Z","timestamp":1713744000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,4,22]],"date-time":"2024-04-22T00:00:00Z","timestamp":1713744000000},"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":["AF 1526513"],"award-info":[{"award-number":["AF 1526513"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS 1723003"],"award-info":[{"award-number":["DMS 1723003"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF 1740761"],"award-info":[{"award-number":["CCF 1740761"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF 1839358"],"award-info":[{"award-number":["CCF 1839358"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006221","name":"United States - Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["2020124"],"award-info":[{"award-number":["2020124"]}],"id":[{"id":"10.13039\/100006221","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2024,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980\u00a0s with Vietoris\u2013Rips Persistent Homology. For given integers <jats:inline-formula><jats:alternatives><jats:tex-math>$$k\\ge 0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$n\\ge 1$$<\/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>\u2265<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> we consider the dimension <jats:italic>k<\/jats:italic> Vietoris\u2013Rips persistence diagrams of <jats:italic>all<\/jats:italic> subsets of a given metric space with cardinality at most <jats:italic>n<\/jats:italic>. We call these invariants <jats:italic>persistence sets<\/jats:italic> and denote them as <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\textbf{D}}_{n,k}^{\\textrm{VR}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msubsup>\n                    <mml:mi>D<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>k<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mtext>VR<\/mml:mtext>\n                  <\/mml:msubsup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We first point out that this family encompasses the usual Vietoris\u2013Rips diagrams. We then establish that (1) for certain range of values of the parameters <jats:italic>n<\/jats:italic> and <jats:italic>k<\/jats:italic>, computing these invariants is significantly more efficient than computing the usual Vietoris\u2013Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris\u2013Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris\u2013Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy\u2019s inequality. We also identify a rich family of metric graphs for which <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\textbf{D}}_{4,1}^{\\textrm{VR}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msubsup>\n                    <mml:mi>D<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mn>4<\/mml:mn>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:mrow>\n                    <mml:mtext>VR<\/mml:mtext>\n                  <\/mml:msubsup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris\u2013Rips persistence diagrams using Mayer\u2013Vietoris sequences. These yield a geometric algorithm for computing the Vietoris\u2013Rips persistence diagram of a space <jats:italic>X<\/jats:italic> with cardinality <jats:inline-formula><jats:alternatives><jats:tex-math>$$2k+2$$<\/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>k<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction.<\/jats:p>","DOI":"10.1007\/s00454-024-00634-0","type":"journal-article","created":{"date-parts":[[2024,4,22]],"date-time":"2024-04-22T20:29:32Z","timestamp":1713817772000},"page":"91-180","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Curvature Sets Over Persistence Diagrams"],"prefix":"10.1007","volume":"72","author":[{"given":"Mario","family":"G\u00f3mez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8409-0549","authenticated-orcid":false,"given":"Facundo","family":"M\u00e9moli","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,4,22]]},"reference":[{"issue":"1","key":"634_CR1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2140\/pjm.2017.290.1","volume":"290","author":"M Adamaszek","year":"2017","unstructured":"Adamaszek, M., Adams, H.: The Vietoris\u2013Rips complexes of a circle. Pac. J. Math. 290(1), 1\u201340 (2017)","journal-title":"Pac. J. Math."},{"issue":"3","key":"634_CR2","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/s41468-020-00054-y","volume":"4","author":"M Adamaszek","year":"2020","unstructured":"Adamaszek, M., Adams, H., Gasparovic, E., Gommel, M., Purvine, E., Sazdanovic, R., Wang, B., Wang, Y., Ziegelmeier, L.: On homotopy types of Vietoris\u2013Rips complexes of metric gluings. J. Appl. Comput. Topol. 4(3), 425\u2013454 (2020)","journal-title":"J. Appl. Comput. Topol."},{"key":"634_CR3","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1016\/j.dam.2014.04.006","volume":"173","author":"M Adamaszek","year":"2014","unstructured":"Adamaszek, M.: Extremal problems related to Betti numbers of flag complexes. Discrete Appl. Math. 173, 8\u201315 (2014)","journal-title":"Discrete Appl. Math."},{"issue":"2","key":"634_CR4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3185466","volume":"14","author":"PK Agarwal","year":"2018","unstructured":"Agarwal, P.K., Fox, K., Nath, A., Sidiropoulos, A., Wang, Y.: Computing the Gromov\u2013Hausdorff distance for metric trees. ACM Trans Algorithms (TALG) 14(2), 1\u201320 (2018)","journal-title":"ACM Trans Algorithms (TALG)"},{"key":"634_CR5","unstructured":"Alipour, M.: Emd (earth movers distance) mex interface. https:\/\/www.mathworks.com\/matlabcentral\/fileexchange\/12936-emd-earth-movers-distance-mex-interface (2023)"},{"key":"634_CR6","doi-asserted-by":"crossref","unstructured":"Alvarez-Melis, D., Jaakkola, T.S.: Gromov-Wasserstein alignment of word embedding spaces. arXiv preprint arXiv:1809.00013 (2018)","DOI":"10.18653\/v1\/D18-1214"},{"key":"634_CR7","unstructured":"Alman, J., Williams, V.V.: A refined laser method and faster matrix multiplication. arXiv preprint arxiv:2010.05846 (2020)"},{"key":"634_CR8","unstructured":"Bauer, U.: Ripser: efficient computation of Vietoris-Rips persistence barcodes, 2019. Software available at http:\/\/ripser.org\/"},{"key":"634_CR9","volume-title":"Numerical Geometry of Non-rigid Shapes","author":"AM Bronstein","year":"2008","unstructured":"Bronstein, A.M., Bronstein, M., Bronstein, M.M., Kimmel, R.: Numerical Geometry of Non-rigid Shapes. Springer, New York (2008)"},{"key":"634_CR10","series-title":"Graduate Studies in Mathematics","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/033","volume-title":"A Course in Metric Geometry","author":"D Burago","year":"2001","unstructured":"Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Socitey, Providence (2001)"},{"key":"634_CR11","unstructured":"Blumberg, A.J., Carriere, M., Mandell, M.A., Rabadan, R., Villar, S.: MREC: a fast and versatile framework for aligning and matching point clouds with applications to single cell molecular data. arXiv preprint arXiv:2001.01666 (2020)"},{"issue":"1","key":"634_CR12","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/0001-8708(92)90061-O","volume":"92","author":"H-J Bandelt","year":"1992","unstructured":"Bandelt, H.-J., Dress, Andreas W M.: A canonical decomposition theory for metrics on a finite set. Adv. Math. 92(1), 47\u2013105 (1992)","journal-title":"Adv. Math."},{"issue":"2","key":"634_CR13","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1017\/S0017089509004984","volume":"51","author":"SM Buckley","year":"2009","unstructured":"Buckley, S.M., Falk, K., Wraith, D.J.: Ptolemaic spaces and CAT(0). Glasg. Math. J. 51(2), 301\u2013314 (2009)","journal-title":"Glasg. Math. J."},{"key":"634_CR14","unstructured":"Blumberg, A.J., Gal, I., Mandell, M.A., Pancia, M.: Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces. arXiv preprint arXiv:1206.4581 (2012)"},{"issue":"4","key":"634_CR15","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1007\/s10208-014-9201-4","volume":"14","author":"AJ Blumberg","year":"2014","unstructured":"Blumberg, A.J., Gal, I., Mandell, M.A., Pancia, M.: Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces. Found. Comput. Math. 14(4), 745\u2013789 (2014)","journal-title":"Found. Comput. Math."},{"issue":"2","key":"634_CR16","doi-asserted-by":"crossref","DOI":"10.1088\/1361-6420\/ab4ac0","volume":"36","author":"P Bubenik","year":"2020","unstructured":"Bubenik, P., Hull, M., Patel, D., Whittle, B.: Persistent homology detects curvature. Inverse Probl. 36(2), 025008 (2020)","journal-title":"Inverse Probl."},{"issue":"4","key":"634_CR17","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1016\/S0196-8858(03)00101-5","volume":"32","author":"M Boutin","year":"2004","unstructured":"Boutin, M., Kemper, G.: On reconstructing $$n$$-point configurations from the distribution of distances or areas. Adv. Appl. Math. 32(4), 709\u2013735 (2004)","journal-title":"Adv. Appl. Math."},{"key":"634_CR18","unstructured":"Bauer, U., Masood, T.Bi. Giunti, B., Houry, G., Kerber, M., Rathod, A.: Keeping it sparse: Computing persistent homology revised. arXiv preprint arxiv:2211.09075 (2022)"},{"key":"634_CR19","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1017\/S0962492914000051","volume":"23","author":"G Carlsson","year":"2014","unstructured":"Carlsson, G.: Topological pattern recognition for point cloud data. Acta Numerica 23, 289\u2013368 (2014)","journal-title":"Acta Numerica"},{"key":"634_CR20","doi-asserted-by":"crossref","DOI":"10.1016\/j.topol.2020.107254","volume":"279","author":"S Chowdhury","year":"2020","unstructured":"Chowdhury, S., Clause, N., M\u00e9moli, F., S\u00e1nchez, J.\u00c1., Wellner, Z.: New families of stable simplicial filtration functors. Topol. Appl. 279, 107254 (2020)","journal-title":"Topol. Appl."},{"issue":"46","key":"634_CR21","doi-asserted-by":"crossref","first-page":"18566","DOI":"10.1073\/pnas.1313480110","volume":"110","author":"JM Chan","year":"2013","unstructured":"Chan, J.M., Carlsson, G., Rabadan, R.: Topology of viral evolution. Proc. Nat. Acad. Sci. 110(46), 18566\u201318571 (2013)","journal-title":"Proc. Nat. Acad. Sci."},{"issue":"5","key":"634_CR22","doi-asserted-by":"crossref","first-page":"1393","DOI":"10.1111\/j.1467-8659.2009.01516.x","volume":"28","author":"F Chazal","year":"2009","unstructured":"Chazal, F., Cohen-Steiner, D., Guibas, L.J., M\u00e9moli, F., Oudot, S.Y.: Gromov\u2013Hausdorff stable signatures for shapes using persistence. Comput. Graph. Forum 28(5), 1393\u20131403 (2009)","journal-title":"Comput. Graph. Forum"},{"issue":"4","key":"634_CR23","doi-asserted-by":"crossref","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."},{"key":"634_CR24","unstructured":"Chazal, F., Fasy, B., Lecci, F., Michel, B., Rinaldo, A., Wasserman, L.: Subsampling methods for persistent homology. arXiv preprint arXiv:1406.1901 (2014)"},{"key":"634_CR25","unstructured":"Chazal, F., Fasy, B., Lecci, F., Michel, B., Rinaldo, A., Wasserman, L.: Subsampling methods for persistent homology. In International Conference on Machine Learning, pp. 2143\u20132151. PMLR (2015)"},{"key":"634_CR26","unstructured":"Carlsson, G., M\u00e9moli, F.: Persistent clustering and a theorem of J. Kleinberg. arXiv preprint arXiv:0808.2241 (2008)"},{"key":"634_CR27","first-page":"1425","volume":"11","author":"G Carlsson","year":"2010","unstructured":"Carlsson, G., M\u00e9moli, F.: Characterization, stability and convergence of hierarchical clustering methods. J. Mach. Learn. Res. 11, 1425\u20131470 (2010)","journal-title":"J. Mach. Learn. Res."},{"issue":"2","key":"634_CR28","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1023\/A:1007992709392","volume":"26","author":"E Calabi","year":"1998","unstructured":"Calabi, E., Olver, P.J., Shakiban, C., Tannenbaum, A., Haker, S.: Differential and numerically invariant signature curves applied to object recognition. Int. J. Comput. Vis. 26(2), 107\u2013135 (1998)","journal-title":"Int. J. Comput. Vis."},{"issue":"1","key":"634_CR29","doi-asserted-by":"crossref","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. Geom. 37(1), 103\u2013120 (2007)","journal-title":"Discrete Comput. Geom."},{"issue":"3","key":"634_CR30","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/0001-8708(84)90029-X","volume":"53","author":"AWM Dress","year":"1984","unstructured":"Dress, A.W.M.: Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces. Adv. Math. 53(3), 321\u2013402 (1984)","journal-title":"Adv. Math."},{"key":"634_CR31","series-title":"Mathematical Surveys and Monographs","doi-asserted-by":"crossref","DOI":"10.1090\/surv\/038","volume-title":"Analysis of and on Uniformly Rectifiable Sets","author":"G David","year":"1993","unstructured":"David, G., Semmes, S.: Analysis of and on Uniformly Rectifiable Sets. Mathematical Surveys and Monographs, American Mathematical Society, Providence (1993)"},{"key":"634_CR32","doi-asserted-by":"crossref","unstructured":"Demetci, P., Santorella, R., Sandstede, B.: William Stafford Noble, and Ritambhara Singh. Gromov-Wasserstein optimal transport to align single-cell multi-omics data, BioRxiv (2020)","DOI":"10.1101\/2020.04.28.066787"},{"key":"634_CR33","unstructured":"Eastwood, P., Ellison, A.M., G\u00f3mez, M., M\u00e9moli, F.: Homology groups of the curvature sets of $${S}^1$$. arXiv preprint arxiv:2209.04674 (2022)"},{"key":"634_CR34","volume-title":"Computational Topology: An Introduction","author":"H Edelsbrunner","year":"2010","unstructured":"Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence (2010)"},{"key":"634_CR35","unstructured":"Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Proceeding of 41st IEEE Symposium on Foundations of Computer Science pp 454\u2013463 (2000)"},{"key":"634_CR36","series-title":"Pure and Applied Mathematics","volume-title":"Real Analysis: Modern Techniques and Their Applications","author":"GB Folland","year":"1999","unstructured":"Folland, G.B.: Real Analysis: Modern Techniques and Their Applications. Pure and Applied Mathematics, 2nd edn. Wiley, New York (1999)","edition":"2"},{"issue":"3","key":"634_CR37","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1017\/S0004972700028574","volume":"42","author":"P Frosini","year":"1990","unstructured":"Frosini, P.: A distance for similarity classes of submanifolds of Euclidean space. Bull. Austral. Math. Soc. 42(3), 407\u2013416 (1990)","journal-title":"Bull. Austral. Math. Soc."},{"key":"634_CR38","unstructured":"Frosini, P.: Omotopie e invarianti metrici per sottovarieta di spazi euclidei (teoria della taglia). PhD thesis, University of Florence (1990)"},{"issue":"2","key":"634_CR39","first-page":"271","volume":"47","author":"P Frosini","year":"1999","unstructured":"Frosini, P.: Metric homotopies. Atti Sem. Mat. Fis. Univ. Modena 47(2), 271\u2013292 (1999)","journal-title":"Atti Sem. Mat. Fis. Univ. Modena"},{"key":"634_CR40","doi-asserted-by":"crossref","unstructured":"Giunti, B., Houry, G., Kerber, M.: Average complexity of matrix reduction for clique filtrations. arXiv preprint arxiv:2111.02125 (2021)","DOI":"10.1145\/3476446.3535474"},{"issue":"1","key":"634_CR41","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1090\/S0273-0979-07-01191-3","volume":"45","author":"R Ghrist","year":"2008","unstructured":"Ghrist, R.: Barcodes: the persistent topology of data. Bull. Am. Math. Soc. 45(1), 61 (2008)","journal-title":"Bull. Am. Math. Soc."},{"key":"634_CR42","unstructured":"G\u00f3mez, M., M\u00e9moli, F.: Github repo for: curvature sets over persistence diagrams (2021). https:\/\/github.com\/ndag\/persistence-curv-sets"},{"key":"634_CR43","unstructured":"G\u00f3mez, M., M\u00e9moli, F.: Curvature sets over persistence diagrams. arXiv preprint arxiv:2103.04470 (2021)"},{"key":"634_CR44","unstructured":"Gomez Flores, M. R. Curvature Sets and Persistent Homology [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http:\/\/rave.ohiolink.edu\/etdc\/view?acc_num=osu1689852191600607 (2023)"},{"key":"634_CR45","series-title":"Mathematical Sciences Research Institute Publications","volume-title":"Hyperbolic Groups. In Essays in Group Theory","author":"M Gromov","year":"1987","unstructured":"Gromov, M.: Hyperbolic Groups. In Essays in Group Theory. Mathematical Sciences Research Institute Publications, Springer, New York (1987)"},{"key":"634_CR46","unstructured":"Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Modern Birkh\u00e4user Classics, Birkh\u00e4user Boston Inc, Boston (2007)"},{"key":"634_CR47","first-page":"175","volume-title":"On the Vietoris\u2013Rips Complexes and a Cohomology Theory for Metric Spaces","author":"J-C Hausmann","year":"1996","unstructured":"Hausmann, J.-C.: On the Vietoris\u2013Rips Complexes and a Cohomology Theory for Metric Spaces, pp. 175\u2013188. Princeton University Press, Princeton (1996)"},{"issue":"6","key":"634_CR48","doi-asserted-by":"crossref","first-page":"1658","DOI":"10.1016\/j.disc.2008.02.037","volume":"309","author":"M Kahle","year":"2009","unstructured":"Kahle, M.: Topology of random clique complexes. Discrete Math. 309(6), 1658\u20131671 (2009)","journal-title":"Discrete Math."},{"issue":"3","key":"634_CR49","doi-asserted-by":"crossref","first-page":"161","DOI":"10.4064\/fm-137-3-161-175","volume":"137","author":"M Katz","year":"1991","unstructured":"Katz, M.: On neighborhoods of the Kuratowski imbedding beyond the first extremum of the diameter functional. Fund. Math. 137(3), 161\u2013175 (1991)","journal-title":"Fund. Math."},{"key":"634_CR50","doi-asserted-by":"crossref","DOI":"10.1016\/j.commatsci.2020.110144","volume":"188","author":"S Kawano","year":"2021","unstructured":"Kawano, S., Mason, J.K.: Classification of atomic environments via the Gromov\u2013Wasserstein distance. Comput. Mater. Sci. 188, 110144 (2021)","journal-title":"Comput. Mater. Sci."},{"key":"634_CR51","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3064175","volume":"22","author":"M Kerber","year":"2017","unstructured":"Kerber, M., Morozov, D., Nigmetov, A.: Geometry helps to compare persistence diagrams. J. Exp. Algorithmics (JEA) 22, 1\u201320 (2017)","journal-title":"J. Exp. Algorithmics (JEA)"},{"key":"634_CR52","unstructured":"Lim, S., Memoli, F., Okutan, O.B.: Vietoris-Rips persistent homology, injective metric spaces, and the filling radius. Algebraic Geom. Topol. (2022)"},{"key":"634_CR53","unstructured":"Lutz, F.H.: Triangulated manifolds with few vertices: combinatorial manifolds (2005)"},{"key":"634_CR54","unstructured":"M\u00e9moli, F.: Estimation of distance functions and geodesics and its use for shape comparison and alignment: theoretical and computational results. PhD thesis, Electrical and Computer Engineering Department, University of Minnesota (2005)"},{"key":"634_CR55","unstructured":"M\u00e9moli, F.: On the use of Gromov\u2013Hausdorff distances for shape comparison. In: Proceedings of Point Based Graphics 2007, Prague (2007)"},{"issue":"4","key":"634_CR56","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/s10208-011-9093-5","volume":"11","author":"F M\u00e9moli","year":"2011","unstructured":"M\u00e9moli, F.: Gromov\u2013Wasserstein distances and the metric approach to object matching. Found. Comput. Math. 11(4), 417\u2013487 (2011)","journal-title":"Found. Comput. Math."},{"key":"634_CR57","unstructured":"M\u00e9moli, F.: Curvature sets over persistence diagrams, 2012. Banff 2012: http:\/\/webfiles.birs.ca\/events\/2012\/5-day-workshops\/12w5081\/videos\/watch\/201210161051-Memoli.html"},{"issue":"2","key":"634_CR58","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1007\/s00454-012-9406-8","volume":"48","author":"F M\u00e9moli","year":"2012","unstructured":"M\u00e9moli, F.: Some properties of Gromov\u2013Hausdorff distances. Discrete Comput. Geom. 48(2), 416\u2013440 (2012)","journal-title":"Discrete Comput. Geom."},{"key":"634_CR59","unstructured":"M\u00e9moli, F.: Curvature sets over persistence diagrams, 2013. ACAT (2013). Bremen: https:\/\/www.alta.uni-bremen.de\/ACAT13\/ACAT13_abstracts.pdf"},{"key":"634_CR60","unstructured":"M\u00e9moli, F.: Curvature sets over persistence diagrams, (2013). Bedlewo 2013: http:\/\/bcc.impan.pl\/13AppTop\/"},{"key":"634_CR61","unstructured":"M\u00e9moli, F.: Curvature sets over persistence diagrams, (2014). IMA 2014: https:\/\/www.ima.umn.edu\/2013-2014\/W10.7-11.13\/14513"},{"key":"634_CR62","unstructured":"M\u00e9moli, F.: Curvature sets over persistence diagrams, (2014). SAMSI 2014: https:\/\/www.samsi.info\/programs-and-activities\/research-workshops\/2013-14-ldhd-topological-data-analysis-february-3-7-2014\/"},{"key":"634_CR63","unstructured":"M\u00e9moli, F.: Curvature sets over persistence diagrams, (2014). SAMSI 2014: https:\/\/people.math.osu.edu\/memolitechera.1\/talks\/talk-dgh-rips.pdf"},{"key":"634_CR64","unstructured":"M\u00e9moli, F.: A distance between filtered spaces via tripods. arXiv preprint arXiv:1704.03965 (2017)"},{"key":"634_CR65","doi-asserted-by":"crossref","unstructured":"Milosavljevi\u0107, N., Morozov, D., Skraba, P.: Zigzag persistent homology in matrix multiplication time. In Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry, SoCG \u201911, pp 216\u2013225, New York (2011). Association for Computing Machinery","DOI":"10.1145\/1998196.1998229"},{"issue":"4","key":"634_CR66","doi-asserted-by":"crossref","first-page":"943","DOI":"10.1111\/sapm.12526","volume":"149","author":"F M\u00e9moli","year":"2022","unstructured":"M\u00e9moli, F., Needham, T.: Distance distributions and inverse problems for metric measure spaces. Stud. Appl. Math. 149(4), 943\u20131001 (2022)","journal-title":"Stud. Appl. Math."},{"key":"634_CR67","unstructured":"Memoli, F., Okutan, O.B., Wang, Q.: Metric graph approximations of geodesic spaces. arXiv preprint arXiv:1809.05566 (2018)"},{"key":"634_CR68","unstructured":"M\u00e9moli, F., Pinto, G.V.F.: Motivic clustering schemes for directed graphs. arXiv preprint arXiv:2001.00278 (2020)"},{"key":"634_CR69","doi-asserted-by":"crossref","unstructured":"M\u00e9moli, F., Sapiro, G.: Comparing point clouds. In: SGP \u201904: Proceedings of the 2004 Eurographics\/ACM SIGGRAPH symposium on Geometry processing, pp. 32\u201340, New York (2004). ACM","DOI":"10.1145\/1057432.1057436"},{"issue":"3","key":"634_CR70","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s10208-004-0145-y","volume":"5","author":"F M\u00e9moli","year":"2005","unstructured":"M\u00e9moli, F., Sapiro, G.: A theoretical and computational framework for isometry invariant recognition of point cloud data. Found. Comput. Math. 5(3), 313\u2013347 (2005)","journal-title":"Found. Comput. Math."},{"key":"634_CR71","unstructured":"M\u00e9moli, F., Smith, Z., Wan, Z.: Gromov\u2013Hausdorff distances on $$ p $$-metric spaces and ultrametric spaces. arXiv preprint arXiv:1912.00564 (2019)"},{"key":"634_CR72","unstructured":"Mugnolo, D.: What is actually a metric graph? arXiv:1912.07549 (2019)"},{"key":"634_CR73","unstructured":"M\u00e9moli, F., Zhou, L.: Persistent homotopy groups of metric spaces. arXiv preprint arXiv:1912.12399 (2019)"},{"issue":"1","key":"634_CR74","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s10208001001","volume":"1","author":"PJ Olver","year":"2001","unstructured":"Olver, P.J.: Joint invariant signatures. Found. Comput. Math. 1(1), 3\u201368 (2001)","journal-title":"Found. Comput. Math."},{"issue":"5\u20136","key":"634_CR75","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1561\/2200000073","volume":"11","author":"G Peyr\u00e9","year":"2019","unstructured":"Peyr\u00e9, G., Cuturi, M., et al.: Computational optimal transport: with applications to data science. Found Trends\u00ae Mach. Learn. 11(5\u20136), 355\u2013607 (2019)","journal-title":"Found Trends\u00ae Mach. Learn."},{"key":"634_CR76","unstructured":"Peyr\u00e9, G., Cuturi, M., Solomon, J.: Gromov\u2013Wasserstein averaging of kernel and distance matrices. In: International Conference on Machine Learning, pp. 2664\u20132672. PMLR (2016)"},{"key":"634_CR77","unstructured":"Robins, V.: Towards computing homology from finite approximations. In: Topology Proceedings 1999 (1999)"},{"issue":"4","key":"634_CR78","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1007\/s00454-017-9889-4","volume":"57","author":"F Schmiedl","year":"2017","unstructured":"Schmiedl, F.: Computational aspects of the Gromov\u2013Hausdorff distance and its application in non-rigid shape matching. Discrete Comput. Geom. 57(4), 854\u2013880 (2017)","journal-title":"Discrete Comput. Geom."},{"issue":"8","key":"634_CR79","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1167\/8.8.11","volume":"8","author":"G Singh","year":"2008","unstructured":"Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. J. Vis. 8(8), 11\u201311 (2008)","journal-title":"J. Vis."},{"issue":"3","key":"634_CR80","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1145\/1015706.1015736","volume":"23","author":"RW Sumner","year":"2004","unstructured":"Sumner, R.W., Popovi\u0107, J.: Deformation transfer for triangle meshes. ACM Trans. Graph. 23(3), 399\u2013405 (2004)","journal-title":"ACM Trans. Graph."},{"key":"634_CR81","unstructured":"Speagle, J.S.: A conceptual introduction to Markov chain monte Carlo methods. arXiv preprint arxiv:1909.12313 (2020)"},{"key":"634_CR82","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0024-3795(82)90001-5","volume":"44","author":"JMS Sim\u00f5es-Pereira","year":"1982","unstructured":"Sim\u00f5es-Pereira, J.M.S., Zamfirescu, C.M.: Submatrices of non-tree-realizable distance matrices. Linear Algebra Appl. 44, 1\u201317 (1982)","journal-title":"Linear Algebra Appl."},{"key":"634_CR83","unstructured":"Solomon, E., Wagner, A., Bendich, P.: From geometry to topology: Inverse theorems for distributed persistence. arXiv preprint arXiv:2101.12288 (2021)"},{"issue":"3","key":"634_CR84","doi-asserted-by":"crossref","first-page":"817","DOI":"10.2140\/pjm.1970.34.817","volume":"34","author":"JE Valentine","year":"1970","unstructured":"Valentine, J.E.: An analogue of Ptolemy\u2019s theorem and its converse in hyperbolic geometry. Pac. J. Math. 34(3), 817\u2013825 (1970)","journal-title":"Pac. J. Math."},{"issue":"1","key":"634_CR85","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1080\/00029890.1970.11992415","volume":"77","author":"JE Valentine","year":"1970","unstructured":"Valentine, J.E.: An analogue of Ptolemy\u2019s theorem in spherical geometry. Am. Math. Month. 77(1), 47\u201351 (1970)","journal-title":"Am. Math. Month."},{"issue":"9","key":"634_CR86","doi-asserted-by":"crossref","first-page":"212","DOI":"10.3390\/a13090212","volume":"13","author":"T Vayer","year":"2020","unstructured":"Vayer, T., Chapel, L., Flamary, R., Tavenard, R., Courty, N.: Fused Gromov\u2013Wasserstein distance for structured objects. Algorithms 13(9), 212 (2020)","journal-title":"Algorithms"},{"key":"634_CR87","series-title":"Graduate Studies in Mathematics","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/058","volume-title":"Topics in Optimal Transportation","author":"C Villani","year":"2003","unstructured":"Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, American Mathematical Society, Providence (2003)"},{"issue":"01","key":"634_CR88","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1142\/S1793525319500444","volume":"12","author":"\u017d Virk","year":"2020","unstructured":"Virk, \u017d: 1-dimensional intrinsic persistence of geodesic spaces. J. Topol. Anal. 12(01), 169\u2013207 (2020)","journal-title":"J. Topol. Anal."},{"issue":"1","key":"634_CR89","first-page":"36","volume":"58","author":"S Weinberger","year":"2011","unstructured":"Weinberger, S.: What is... persistent homology? Not. AMS 58(1), 36\u201339 (2011)","journal-title":"Not. AMS"},{"key":"634_CR90","doi-asserted-by":"crossref","unstructured":"Zomorodian, A., Carlsson, G.: Computing persistent homology. In: SCG \u201904: Proceedings of the twentieth annual symposium on Computational geometry, pp. 347\u2013356, New York (2004). ACM","DOI":"10.1145\/997817.997870"},{"key":"634_CR91","unstructured":"Zhang, S., Xiao, M., Wang, H.: GPU-accelerated computation of Vietoris\u2013Rips persistence barcodes. In: 36th International Symposium on Computational Geometry (SoCG 2020). Schloss Dagstuhl\u2013Leibniz\u2013Zentrum f\u00fcr Informatik (2020)"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-024-00634-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-024-00634-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-024-00634-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,1]],"date-time":"2024-06-01T21:03:08Z","timestamp":1717275788000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-024-00634-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,22]]},"references-count":91,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,7]]}},"alternative-id":["634"],"URL":"https:\/\/doi.org\/10.1007\/s00454-024-00634-0","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,22]]},"assertion":[{"value":"7 April 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 January 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 January 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 April 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}