{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T08:06:51Z","timestamp":1773475611046,"version":"3.50.1"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T00:00:00Z","timestamp":1593129600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T00:00:00Z","timestamp":1593129600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"name":"National Science Foundation","award":["DMS - 1764406"],"award-info":[{"award-number":["DMS - 1764406"]}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["594594"],"award-info":[{"award-number":["594594"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"crossref","award":["W911NF1810307"],"award-info":[{"award-number":["W911NF1810307"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Appl. and Comput. Topology"],"published-print":{"date-parts":[[2020,9]]},"DOI":"10.1007\/s41468-020-00056-w","type":"journal-article","created":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T11:45:05Z","timestamp":1593171905000},"page":"339-351","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Embeddings of persistence diagrams into Hilbert spaces"],"prefix":"10.1007","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5262-2133","authenticated-orcid":false,"given":"Peter","family":"Bubenik","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5961-7852","authenticated-orcid":false,"given":"Alexander","family":"Wagner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,6,26]]},"reference":[{"key":"56_CR1","unstructured":"Bell, G., Lawson, A., Pritchard, C.N., Yasaki, D.: The space of persistence diagrams fails to have Yu\u2019s property A. arXiv e-prints arXiv:1902.02288 (2019)"},{"key":"56_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1128-0","volume-title":"Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions, Graduate Texts in Mathematics","author":"C Berg","year":"1984","unstructured":"Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions, Graduate Texts in Mathematics, vol. 100. Springer, New York (1984). https:\/\/doi.org\/10.1007\/978-1-4612-1128-0"},{"issue":"4","key":"56_CR3","doi-asserted-by":"publisher","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). https:\/\/doi.org\/10.1007\/s10208-014-9201-4","journal-title":"Found. Comput. Math."},{"issue":"3\u20134","key":"56_CR4","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1007\/s41468-018-0022-4","volume":"2","author":"P Bubenik","year":"2018","unstructured":"Bubenik, P., Vergili, T.: Topological spaces of persistence modules and their properties. J. Appl. Comput. Topol. 2(3\u20134), 233\u2013269 (2018). https:\/\/doi.org\/10.1007\/s41468-018-0022-4","journal-title":"J. Appl. Comput. Topol."},{"key":"56_CR5","unstructured":"Carri\u00e8re, M., Bauer, U.: On the metric distortion of embedding persistence diagrams into separable Hilbert spaces. In: 35th International Symposium on Computational Geometry, LIPIcs. Leibniz Int. Proc. Inform., vol 129, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, pp Art. No. 21, 15, https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=3968607 (2019)"},{"key":"56_CR6","doi-asserted-by":"publisher","unstructured":"Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.Y.: Proximity of persistence modules and their diagrams. In: Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, ACM, New York, NY, USA, SCG \u201909, pp 237\u2013246. https:\/\/doi.org\/10.1145\/1542362.1542407 (2009)","DOI":"10.1145\/1542362.1542407"},{"issue":"1","key":"56_CR7","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. Geom. 37(1), 103\u2013120 (2007). https:\/\/doi.org\/10.1007\/s00454-006-1276-5","journal-title":"Discrete Comput. Geom."},{"issue":"2","key":"56_CR8","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1007\/s10208-010-9060-6","volume":"10","author":"D Cohen-Steiner","year":"2010","unstructured":"Cohen-Steiner, D., Edelsbrunner, H., Harer, J., Mileyko, Y.: Lipschitz functions have $$L_p$$-stable persistence. Found. Comput. Math. 10(2), 127\u2013139 (2010). https:\/\/doi.org\/10.1007\/s10208-010-9060-6","journal-title":"Found. Comput. Math."},{"issue":"1","key":"56_CR9","doi-asserted-by":"publisher","first-page":"60","DOI":"10.4153\/CMB-2002-006-9","volume":"45","author":"AN Dranishnikov","year":"2002","unstructured":"Dranishnikov, A.N., Gong, G., Lafforgue, V., Yu, G.: Uniform embeddings into Hilbert space and a question of Gromov. Can. Math. Bull. 45(1), 60\u201370 (2002). https:\/\/doi.org\/10.4153\/CMB-2002-006-9","journal-title":"Can. Math. Bull."},{"key":"56_CR10","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/BF02589550","volume":"8","author":"P Enflo","year":"1969","unstructured":"Enflo, P.: On a problem of Smirnov. Ark. Mat. 8, 107\u2013109 (1969). https:\/\/doi.org\/10.1007\/BF02589550","journal-title":"Ark. Mat."},{"key":"56_CR11","unstructured":"Gromov, M.: Asymptotic invariants of infinite groups. In: Geometric Group Theory, Vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser., vol. 182, pp. 1\u2013295. Cambridge University Press, Cambridge (1993)"},{"issue":"1","key":"56_CR12","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1307\/mmj\/1029005619","volume":"44","author":"CJ Lennard","year":"1997","unstructured":"Lennard, C.J., Tonge, A.M., Weston, A.: Generalized roundness and negative type. Michigan Math. J. 44(1), 37\u201345 (1997). https:\/\/doi.org\/10.1307\/mmj\/1029005619","journal-title":"Michigan Math. J."},{"key":"56_CR13","doi-asserted-by":"publisher","unstructured":"Mileyko, Y., Mukherjee, S., Harer, J.: Probability measures on the space of persistence diagrams. Inverse Problems 27(12):124007, 22. https:\/\/doi.org\/10.1088\/0266-5611\/27\/12\/124007 (2011)","DOI":"10.1088\/0266-5611\/27\/12\/124007"},{"key":"56_CR14","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/209","volume-title":"Persistence Theory: From Quiver Representations to Data Analysis, Mathematical Surveys and Monographs","author":"SY Oudot","year":"2015","unstructured":"Oudot, S.Y.: Persistence Theory: From Quiver Representations to Data Analysis, Mathematical Surveys and Monographs, vol. 209. American Mathematical Society, Providence (2015). https:\/\/doi.org\/10.1090\/surv\/209"},{"key":"56_CR15","doi-asserted-by":"publisher","unstructured":"Roe, J.: Lectures on coarse geometry, University Lecture Series, vol 31. American Mathematical Society, Providence, RI (2003). https:\/\/doi.org\/10.1090\/ulect\/031","DOI":"10.1090\/ulect\/031"},{"issue":"3","key":"56_CR16","doi-asserted-by":"publisher","first-page":"724","DOI":"10.2307\/1968654","volume":"36","author":"IJ Schoenberg","year":"1935","unstructured":"Schoenberg, I.J.: Remarks to Maurice Fr\u00e9chet\u2019s article. Ann. Math (2) 36(3), 724\u2013732 (1935). https:\/\/doi.org\/10.2307\/1968654","journal-title":"Ann. Math (2)"},{"issue":"3","key":"56_CR17","doi-asserted-by":"publisher","first-page":"522","DOI":"10.2307\/1989894","volume":"44","author":"IJ Schoenberg","year":"1938","unstructured":"Schoenberg, I.J.: Metric spaces and positive definite functions. Trans. Am. Math. Soc. 44(3), 522\u2013536 (1938). https:\/\/doi.org\/10.2307\/1989894","journal-title":"Trans. Am. Math. Soc."},{"key":"56_CR18","volume-title":"Support Vector Machines. Information Science and Statistics","author":"I Steinwart","year":"2008","unstructured":"Steinwart, I., Christmann, A.: Support Vector Machines. Information Science and Statistics. Springer, New York (2008)"},{"key":"56_CR19","doi-asserted-by":"crossref","unstructured":"Turner, K., Spreemann, G.: Same but Different: distance correlations between topological summaries. arXiv e-prints arXiv:1903.01051 (2019)","DOI":"10.1007\/978-3-030-43408-3_18"},{"key":"56_CR20","unstructured":"Wells, J.H., Williams, L.R.: Embeddings and extensions in analysis. Springer, New York, ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84 (1975)"},{"issue":"1","key":"56_CR21","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s002229900032","volume":"139","author":"G Yu","year":"2000","unstructured":"Yu, G.: The coarse Baum\u2013Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent Math. 139(1), 201\u2013240 (2000). https:\/\/doi.org\/10.1007\/s002229900032","journal-title":"Invent Math."}],"container-title":["Journal of Applied and Computational Topology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41468-020-00056-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s41468-020-00056-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41468-020-00056-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,25]],"date-time":"2021-06-25T23:18:20Z","timestamp":1624663100000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s41468-020-00056-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,26]]},"references-count":21,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["56"],"URL":"https:\/\/doi.org\/10.1007\/s41468-020-00056-w","relation":{},"ISSN":["2367-1726","2367-1734"],"issn-type":[{"value":"2367-1726","type":"print"},{"value":"2367-1734","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,26]]},"assertion":[{"value":"25 May 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 June 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 June 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interest"}}]}}