{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T21:41:41Z","timestamp":1784065301251,"version":"3.55.0"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2019,2,21]],"date-time":"2019-02-21T00:00:00Z","timestamp":1550707200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,2,21]],"date-time":"2019-02-21T00:00:00Z","timestamp":1550707200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100009023","name":"Precursory Research for Embryonic Science and Technology","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100009023","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["KAKENHI (26280009)"],"award-info":[{"award-number":["KAKENHI (26280009)"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2020,2]]},"DOI":"10.1007\/s11222-019-09855-3","type":"journal-article","created":{"date-parts":[[2019,2,21]],"date-time":"2019-02-21T02:30:27Z","timestamp":1550716227000},"page":"1-18","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Empirical geodesic graphs and CAT(k) metrics for data analysis"],"prefix":"10.1007","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4567-5883","authenticated-orcid":false,"given":"Kei","family":"Kobayashi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6448-1080","authenticated-orcid":false,"given":"Henry P.","family":"Wynn","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2019,2,21]]},"reference":[{"issue":"2","key":"9855_CR1","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1006\/asle.2001.0025","volume":"1","author":"L Alexander","year":"2000","unstructured":"Alexander, L., Jones, P.: Updated precipitation series for the UK and discussion of recent extremes. Atmos. Sci. Lett. 1(2), 142\u2013150 (2000)","journal-title":"Atmos. Sci. Lett."},{"key":"9855_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5056-2","volume-title":"Differential-Geometrical Methods in Statistics","author":"SI Amari","year":"1985","unstructured":"Amari, S.I.: Differential-Geometrical Methods in Statistics. Springer, New York (1985)"},{"key":"9855_CR3","doi-asserted-by":"publisher","DOI":"10.1090\/mmono\/191","volume-title":"Methods of Information Geometry","author":"SI Amari","year":"2007","unstructured":"Amari, S.I., Nagaoka, H.: Methods of Information Geometry. American Mathematical Society, Providence (2007)"},{"key":"9855_CR4","unstructured":"Arjovsky, M., Chintala, S., Bottou, L.: Wasserstein GAN (2017). arXiv:1701.07875"},{"key":"9855_CR5","series-title":"Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge\/A Series of Modern Surveys in Mathematics","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-56478-4","volume-title":"Information Geometry","author":"N Ay","year":"2017","unstructured":"Ay, N., Jost, J., L\u00ea, H.V., Schwachh\u00f6fer, L.: Information Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge\/A Series of Modern Surveys in Mathematics. Springer, Berlin (2017)"},{"key":"9855_CR6","unstructured":"Bache, K., Lichman, M.: UCI Machine Learning Repository (2013). http:\/\/archive.ics.uci.edu\/ml\/ . Accessed 9 Jan 2015"},{"key":"9855_CR7","doi-asserted-by":"crossref","first-page":"585","DOI":"10.7551\/mitpress\/1120.003.0080","volume-title":"Advances in Neural Information Processing Systems 14","author":"M Belkin","year":"2002","unstructured":"Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Dietterich, T.G., Becker, S., Ghahramani, Z. (eds.) Advances in Neural Information Processing Systems 14, pp. 585\u2013591. MIT Press, Cambridge (2002)"},{"issue":"8","key":"9855_CR8","doi-asserted-by":"publisher","first-page":"1798","DOI":"10.1109\/TPAMI.2013.50","volume":"35","author":"Y Bengio","year":"2013","unstructured":"Bengio, Y., Courville, A., Vincent, P.: Representation learning: a review and new perspectives. IEEE Trans. Pattern Anal. Mach. Intell. 35(8), 1798\u20131828 (2013)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"issue":"1","key":"9855_CR9","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1093\/nar\/28.1.235","volume":"28","author":"HM Berman","year":"2000","unstructured":"Berman, H.M., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T.N., Weissig, H., Shindyalov, I.N., Bourne, P.E.: The protein data bank. Nucleic Acids Res. 28(1), 235\u2013242 (2000)","journal-title":"Nucleic Acids Res."},{"key":"9855_CR10","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139094764","volume-title":"Nonparametric Inference on Manifolds: With Applications to Shape Spaces","author":"A Bhattacharya","year":"2012","unstructured":"Bhattacharya, A., Bhattacharya, R.: Nonparametric Inference on Manifolds: With Applications to Shape Spaces. Cambridge University Press, Cambridge (2012a)"},{"key":"9855_CR11","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139094764","volume-title":"Nonparametric inference on manifolds: with applications to shape spaces","author":"A Bhattacharya","year":"2012","unstructured":"Bhattacharya, A., Bhattacharya, R.: Nonparametric inference on manifolds: with applications to shape spaces, vol. 2. Cambridge University Press, Cambridge (2012b)"},{"issue":"4","key":"9855_CR12","doi-asserted-by":"publisher","first-page":"733","DOI":"10.1006\/aama.2001.0759","volume":"27","author":"LJ Billera","year":"2001","unstructured":"Billera, L.J., Holmes, S.P., Vogtmann, K.: Geometry of the space of phylogenetic trees. Adv. Appl. Math. 27(4), 733\u2013767 (2001)","journal-title":"Adv. Appl. Math."},{"key":"9855_CR13","volume-title":"Metric Spaces of Non-positive Curvature","author":"MR Bridson","year":"2011","unstructured":"Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, vol. 319. Springer, Berlin (2011)"},{"issue":"1\u201317","key":"9855_CR14","first-page":"1","volume":"12","author":"L Cayton","year":"2005","unstructured":"Cayton, L.: Algorithms for manifold learning. Univ. Calif. San Diego Tech. Rep. 12(1\u201317), 1 (2005)","journal-title":"Univ. Calif. San Diego Tech. Rep."},{"key":"9855_CR15","unstructured":"CIESEN, FAO, CIAT: Gridded population of the world, version 3 (GPWV3): population count grid (2005). http:\/\/dx.doi.org\/10.7927\/H4639MPP . Accessed 16 June 2016"},{"key":"9855_CR16","unstructured":"Cuturi, M., Doucet, A.: Fast computation of Wasserstein barycenters. In: International Conference on Machine Learning, jmlr.org, pp. 685\u2013693 (2014)"},{"key":"9855_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/978-3-662-04245-8","volume-title":"Computational Geometry","author":"M De Berg","year":"2000","unstructured":"De Berg, M., Van Kreveld, M., Overmars, M., Schwarzkopf, O.C.: Computational Geometry, pp. 1\u201317. Springer, Berlin (2000)"},{"key":"9855_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/978-3-642-00234-2","volume-title":"Encyclopedia of Distances","author":"MM Deza","year":"2009","unstructured":"Deza, M.M., Deza, E.: Encyclopedia of Distances, pp. 1\u2013583. Springer, Berlin (2009)"},{"key":"9855_CR19","doi-asserted-by":"publisher","DOI":"10.1002\/9781119072492","volume-title":"Statistical Shape Analysis: With Applications in R","author":"IL Dryden","year":"2016","unstructured":"Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis: With Applications in R. Wiley, Hoboken (2016)"},{"issue":"6","key":"9855_CR20","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1145\/367766.368168","volume":"5","author":"RW Floyd","year":"1962","unstructured":"Floyd, R.W.: Algorithm 97: shortest path. Commun. ACM 5(6), 345 (1962)","journal-title":"Commun. ACM"},{"key":"9855_CR21","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1007\/978-1-4613-9586-7_3","volume-title":"Essays in Group Theory","author":"M Gromov","year":"1987","unstructured":"Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory, pp. 75\u2013263. Springer, Berlin (1987)"},{"issue":"2","key":"9855_CR22","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1112\/blms\/16.2.81","volume":"16","author":"DG Kendall","year":"1984","unstructured":"Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81\u2013121 (1984)","journal-title":"Bull. Lond. Math. Soc."},{"key":"9855_CR23","unstructured":"Kendall, M.G., Mor\u00e1n, PA.: Geometrical probability. Technical Report (1963)"},{"key":"9855_CR24","volume-title":"Shape and Shape Theory","author":"DG Kendall","year":"2009","unstructured":"Kendall, D.G., Barden, D., Carne, T.K., Le, H.: Shape and Shape Theory. Wiley, Hoboken (2009)"},{"issue":"2","key":"9855_CR25","doi-asserted-by":"publisher","first-page":"808","DOI":"10.1214\/009053606000000010","volume":"34","author":"F Komaki","year":"2006","unstructured":"Komaki, F.: Shrinkage priors for Bayesian prediction. Ann. Stat. 34(2), 808\u2013819 (2006)","journal-title":"Ann. Stat."},{"issue":"4","key":"9855_CR26","doi-asserted-by":"publisher","first-page":"605","DOI":"10.2748\/tmj\/1325886283","volume":"63","author":"Y Lin","year":"2011","unstructured":"Lin, Y., Lu, L., Yau, S.T.: Ricci curvature of graphs. Tohoku Math. J. 63(4), 605\u2013627 (2011)","journal-title":"Tohoku Math. J."},{"issue":"5","key":"9855_CR27","doi-asserted-by":"publisher","first-page":"732","DOI":"10.1002\/bimj.201300072","volume":"56","author":"JS Marron","year":"2014","unstructured":"Marron, J.S., Alonso, A.M.: Overview of object oriented data analysis. Biom. J. 56(5), 732\u2013753 (2014)","journal-title":"Biom. J."},{"key":"9855_CR28","volume-title":"Tensor Methods in Statistics","author":"P McCullagh","year":"1986","unstructured":"McCullagh, P.: Tensor Methods in Statistics. Courier Dover Publications, Mineola (1986)"},{"issue":"5","key":"9855_CR29","doi-asserted-by":"publisher","first-page":"2716","DOI":"10.1214\/11-AOS915","volume":"39","author":"TMW Nye","year":"2011","unstructured":"Nye, T.M.W.: Principal components analysis in the space of phylogenetic trees. Ann. Stat. 39(5), 2716\u20132739 (2011)","journal-title":"Ann. Stat."},{"key":"9855_CR30","volume-title":"Spatial Tessellations: Concepts and Applications of Voronoi Diagrams","author":"A Okabe","year":"2009","unstructured":"Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, vol. 501. Wiley, Hoboken (2009)"},{"issue":"1","key":"9855_CR31","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1109\/TCBB.2010.3","volume":"8","author":"M Owen","year":"2011","unstructured":"Owen, M., Provan, J.S.: A fast algorithm for computing geodesic distances in tree space. IEEE\/ACM Trans. Comput. Biol. Bioinform. 8(1), 2\u201313 (2011)","journal-title":"IEEE\/ACM Trans. Comput. Biol. Bioinform."},{"key":"9855_CR32","unstructured":"Panaretos, V.M., Zemel, Y.: Statistical aspects of Wasserstein distances (2018). arXiv:1806.05500"},{"key":"9855_CR33","doi-asserted-by":"publisher","DOI":"10.1201\/b18969","volume-title":"Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis","author":"V Patrangenaru","year":"2015","unstructured":"Patrangenaru, V., Ellingson, L.: Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis. CRC Press, Boca Raton (2015)"},{"key":"9855_CR34","unstructured":"Peyr\u00e9, G., Cuturi, M.: Computational optimal transport (2018). arXiv:1803.00567"},{"key":"9855_CR35","volume-title":"Applied Functional Data Analysis: Methods and Case Studies","author":"JO Ramsay","year":"2007","unstructured":"Ramsay, J.O., Silverman, B.W.: Applied Functional Data Analysis: Methods and Case Studies. Springer, Berlin (2007)"},{"key":"9855_CR36","first-page":"119","volume":"4","author":"LK Saul","year":"2003","unstructured":"Saul, L.K.: Think globally, fit locally: unsupervised learning of low dimensional manifolds. J. Mach. Learn. Res. 4, 119\u2013155 (2003)","journal-title":"J. Mach. Learn. Res."},{"issue":"4","key":"9855_CR37","doi-asserted-by":"publisher","first-page":"66:1","DOI":"10.1145\/2766963","volume":"34","author":"J Solomon","year":"2015","unstructured":"Solomon, J., de Goes, F., Peyr\u00e9, G., Cuturi, M., Butscher, A., Nguyen, A., Du, T., Guibas, L.: Convolutional wasserstein distances: efficient optimal transportation on geometric domains. ACM Trans. Gr. 34(4), 66:1\u201366:11 (2015)","journal-title":"ACM Trans. Gr."},{"key":"9855_CR38","series-title":"Springer Series in Statistics","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4939-4020-2","volume-title":"Functional and Shape Data Analysis","author":"A Srivastava","year":"2016","unstructured":"Srivastava, A., Klassen, E.P.: Functional and Shape Data Analysis. Springer Series in Statistics. Springer, Berlin (2016)"},{"issue":"4","key":"9855_CR39","first-page":"1005","volume":"48","author":"A Takatsu","year":"2011","unstructured":"Takatsu, A.: Wasserstein geometry of Gaussian measures. Osaka J. Math. 48(4), 1005\u20131026 (2011)","journal-title":"Osaka J. Math."},{"issue":"3","key":"9855_CR40","doi-asserted-by":"publisher","first-page":"444","DOI":"10.1111\/j.1467-9892.2007.00561.x","volume":"29","author":"F Tanaka","year":"2008","unstructured":"Tanaka, F., Komaki, F.: A superharmonic prior for the autoregressive process of the second-order. J. Time Ser. Anal. 29(3), 444\u2013452 (2008)","journal-title":"J. Time Ser. Anal."},{"issue":"5500","key":"9855_CR41","doi-asserted-by":"publisher","first-page":"2319","DOI":"10.1126\/science.290.5500.2319","volume":"290","author":"JB Tenenbaum","year":"2000","unstructured":"Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319\u20132323 (2000)","journal-title":"Science"},{"issue":"4","key":"9855_CR42","doi-asserted-by":"publisher","first-page":"784","DOI":"10.1137\/1118101","volume":"18","author":"S Vallender","year":"1974","unstructured":"Vallender, S.: Calculation of the Wasserstein distance between probability distributions on the line. Theory Probab. Appl. 18(4), 784\u2013786 (1974)","journal-title":"Theory Probab. Appl."},{"key":"9855_CR43","volume-title":"Optimal Transport: Old and New","author":"C Villani","year":"2008","unstructured":"Villani, C.: Optimal Transport: Old and New. Springer, Berlin (2008)"},{"issue":"5","key":"9855_CR44","doi-asserted-by":"publisher","first-page":"1849","DOI":"10.1214\/009053607000000217","volume":"35","author":"Haonan Wang","year":"2007","unstructured":"Wang, H., Marron, J.S.: Object oriented data analysis: sets of trees (2007). arXiv:0711.3147","journal-title":"The Annals of Statistics"},{"issue":"2","key":"9855_CR45","first-page":"4","volume":"2","author":"L Yang","year":"2006","unstructured":"Yang, L., Jin, R.: Distance metric learning: a comprehensive survey. Mich. State Univ. 2(2), 4 (2006)","journal-title":"Mich. State Univ."}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11222-019-09855-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-019-09855-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-019-09855-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,14]],"date-time":"2024-07-14T23:42:42Z","timestamp":1721000562000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11222-019-09855-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,21]]},"references-count":45,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,2]]}},"alternative-id":["9855"],"URL":"https:\/\/doi.org\/10.1007\/s11222-019-09855-3","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,2,21]]},"assertion":[{"value":"22 March 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 February 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 February 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}