{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T03:01:25Z","timestamp":1781665285010,"version":"3.54.5"},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T00:00:00Z","timestamp":1737331200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T00:00:00Z","timestamp":1737331200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004063","name":"Knut och Alice Wallenbergs Stiftelse","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004063","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004359","name":"Vetenskapsr\u00e5det","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004359","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004270","name":"Royal Institute of Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004270","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":[[2025,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Distances have a ubiquitous role in persistent homology, from the direct comparison of homological representations of data to the definition and optimization of invariants. In this article we introduce a family of parametrized pseudometrics between persistence modules based on the algebraic Wasserstein distance defined by Skraba and Turner, and phrase them in the formalism of noise systems. This is achieved by comparing <jats:italic>p<\/jats:italic>-norms of cokernels (resp. kernels) of monomorphisms (resp. epimorphisms) between persistence modules and corresponding bar-to-bar morphisms, a novel notion that allows us to bridge between algebraic and combinatorial aspects of persistence modules. We use algebraic Wasserstein distances to define invariants, called Wasserstein stable ranks, which are 1-Lipschitz stable with respect to such pseudometrics. We prove a low-rank approximation result for persistence modules which allows us to efficiently compute Wasserstein stable ranks, and we propose an efficient algorithm to compute the interleaving distance between them. Importantly, Wasserstein stable ranks depend on interpretable parameters which can be learnt in a machine learning context. Experimental results illustrate the use of Wasserstein stable ranks on real and artificial data and highlight how such pseudometrics could be useful in data analysis tasks.<\/jats:p>","DOI":"10.1007\/s41468-024-00200-w","type":"journal-article","created":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T03:00:56Z","timestamp":1737342056000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Algebraic Wasserstein distances and stable homological invariants of data"],"prefix":"10.1007","volume":"9","author":[{"given":"Jens","family":"Agerberg","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrea","family":"Guidolin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Isaac","family":"Ren","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martina","family":"Scolamiero","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,1,20]]},"reference":[{"key":"200_CR1","doi-asserted-by":"crossref","DOI":"10.3389\/frai.2021.668302","volume":"4","author":"H Adams","year":"2021","unstructured":"Adams, H., Moy, M.: Topology applied to machine learning: from global to local. Front. Artif. Intell. 4, 668302 (2021)","journal-title":"Front. Artif. Intell."},{"key":"200_CR2","first-page":"1","volume":"18","author":"H Adams","year":"2017","unstructured":"Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Shipman, P., Chepushtanova, S., Hanson, E., Motta, F., Ziegelmeier, L.: Persistence images: a stable vector representation of persistent homology. J. Mach. Learn. Res. 18, 1\u201335 (2017)","journal-title":"J. Mach. Learn. Res."},{"key":"200_CR3","doi-asserted-by":"crossref","DOI":"10.3389\/fams.2021.668046","volume":"7","author":"J Agerberg","year":"2021","unstructured":"Agerberg, J., Ramanujam, R., Scolamiero, M., Chach\u00f3lski, W.: Supervised learning using homology stable rank kernels. Front. Appl. Math. Stat. 7, 668046 (2021)","journal-title":"Front. Appl. Math. Stat."},{"issue":"2","key":"200_CR4","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":"200_CR5","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/978-3-030-43408-3_3","volume-title":"Topological Data Analysis","author":"U Bauer","year":"2020","unstructured":"Bauer, U., Lesnick, M.: Persistence diagrams as diagrams: a categorification of the stability theorem. In: Baas, N.A., Carlsson, G.E., Quick, G., Szymik, M., Thaule, M. (eds.) Topological Data Analysis, pp. 67\u201396. Springer, Cham (2020)"},{"issue":"2","key":"200_CR6","doi-asserted-by":"crossref","first-page":"297","DOI":"10.4310\/HHA.2023.v25.n2.a13","volume":"25","author":"U Bauer","year":"2023","unstructured":"Bauer, U., Schmahl, M.: Lifespan functors and natural dualities in persistent homology. Homol. Homotopy Appl. 25(2), 297\u2013327 (2023)","journal-title":"Homol. Homotopy Appl."},{"issue":"1","key":"200_CR7","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1214\/15-AOAS886","volume":"10","author":"P Bendich","year":"2016","unstructured":"Bendich, P., Marron, J.S., Miller, E., Pieloch, A., Skwerer, S.: Persistent homology analysis of brain artery trees. Ann. Appl. Stat. 10(1), 198 (2016)","journal-title":"Ann. Appl. Stat."},{"issue":"5","key":"200_CR8","doi-asserted-by":"crossref","first-page":"1233","DOI":"10.1007\/s10208-020-09482-9","volume":"21","author":"P Bubenik","year":"2021","unstructured":"Bubenik, P., Mili\u0107evi\u0107, N.: Homological algebra for persistence modules. Found. Comput. Math. 21(5), 1233\u20131278 (2021)","journal-title":"Found. Comput. Math."},{"issue":"6","key":"200_CR9","doi-asserted-by":"crossref","first-page":"1501","DOI":"10.1007\/s10208-014-9229-5","volume":"15","author":"P Bubenik","year":"2015","unstructured":"Bubenik, P., De Silva, V., Scott, J.: Metrics for generalized persistence modules. Found. Comput. Math. 15(6), 1501\u20131531 (2015)","journal-title":"Found. Comput. Math."},{"issue":"2","key":"200_CR10","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/s41468-022-00103-8","volume":"7","author":"P Bubenik","year":"2023","unstructured":"Bubenik, P., Scott, J., Stanley, D.: Exact weights, path metrics, and algebraic Wasserstein distances. J. Appl. Comput. Topol. 7(2), 185\u2013219 (2023)","journal-title":"J. Appl. Comput. Topol."},{"issue":"2","key":"200_CR11","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1016\/j.neurobiolaging.2008.03.022","volume":"31","author":"E Bullitt","year":"2010","unstructured":"Bullitt, E., Zeng, D., Mortamet, B., Ghosh, A., Aylward, S.R., Lin, W., Marks, B.L., Smith, K.: The effects of healthy aging on intracerebral blood vessels visualized by magnetic resonance angiography. Neurobiol. Aging 31(2), 290\u2013300 (2010)","journal-title":"Neurobiol. Aging"},{"key":"200_CR12","unstructured":"Carri\u00e8re, M., Chazal, F., Ike, Y., Lacombe, T., Royer, M., Umeda, Y.: Perslay: A neural network layer for persistence diagrams and new graph topological signatures. In: International Conference on Artificial Intelligence and Statistics, pp. 2786\u20132796. PMLR (2020)"},{"issue":"1","key":"200_CR13","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1137\/19M1243932","volume":"4","author":"W Chach\u00f3lski","year":"2020","unstructured":"Chach\u00f3lski, W., Riihim\u00e4ki, H.: Metrics and stabilization in one parameter persistence. SIAM J. Appl. Algebra Geom. 4(1), 69\u201398 (2020)","journal-title":"SIAM J. Appl. Algebra Geom."},{"key":"200_CR14","series-title":"Springer Briefs in Mathematics","doi-asserted-by":"crossref","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. Springer Briefs in Mathematics, vol. 10. Springer, New York (2016)"},{"key":"200_CR15","doi-asserted-by":"crossref","unstructured":"Chen, C., Edelsbrunner, H.: Diffusion runs low on persistence fast. In: 2011 International Conference on Computer Vision, pp. 423\u2013430. IEEE (2011)","DOI":"10.1109\/ICCV.2011.6126271"},{"key":"200_CR16","unstructured":"Chen, Y.-C., Wang, D., Rinaldo, A., Wasserman, L.: Statistical analysis of persistence intensity functions. arXiv preprint arXiv:1510.02502v1 (2015)"},{"key":"200_CR17","unstructured":"Chubet, O.A., Gardner, K.P., Sheehy, D.R.: A theory of sub-barcodes. arXiv preprint arXiv:2206.10504v1 (2022)"},{"key":"200_CR18","doi-asserted-by":"crossref","unstructured":"Chung, Y.-M., Hu, C.-S., Lawson, A., Smyth, C.: Topological approaches to skin disease image analysis. In: 2018 IEEE International Conference on Big Data (Big Data), pp. 100\u2013105. IEEE (2018)","DOI":"10.1109\/BigData.2018.8622175"},{"key":"200_CR19","doi-asserted-by":"crossref","unstructured":"Cohen-Steiner, D., Edelsbrunner, H., Morozov, D.: Vines and vineyards by updating persistence in linear time. In: Proceedings of the Twenty-second Annual Symposium on Computational Geometry, pp. 119\u2013126 (2006)","DOI":"10.1145\/1137856.1137877"},{"issue":"2","key":"200_CR20","doi-asserted-by":"crossref","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)","journal-title":"Found. Comput. Math."},{"issue":"21","key":"200_CR21","first-page":"583","volume":"33","author":"V de Silva","year":"2018","unstructured":"de Silva, V., Munch, E., Stefanou, A.: Theory of interleavings on categories with a flow. Theory Appl. Categ. 33(21), 583\u2013607 (2018)","journal-title":"Theory Appl. Categ."},{"key":"200_CR22","unstructured":"Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Proceedings 41st Annual Symposium on Foundations of Computer Science, pp. 454\u2013463. IEEE (2000)"},{"key":"200_CR23","unstructured":"G\u00e4fvert, O., Chach\u00f3lski, W.: Stable invariants for multidimensional persistence. arXiv preprint arXiv:1703.03632v3 (2017)"},{"key":"200_CR24","unstructured":"G\u00e4fvert, O.: Topology-based metric learning. Conference Poster. Available: https:\/\/people.kth.se\/~oliverg\/TAGS_poster.pdf (2018)"},{"key":"200_CR25","doi-asserted-by":"crossref","unstructured":"Garin, A., Tauzin, G.: A topological \u201creading\u201d lesson: classification of MNIST using TDA. In: 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA), pp. 1551\u20131556. IEEE (2019)","DOI":"10.1109\/ICMLA.2019.00256"},{"issue":"5","key":"200_CR26","doi-asserted-by":"crossref","first-page":"0217413","DOI":"10.1371\/journal.pone.0217413","volume":"14","author":"K Garside","year":"2019","unstructured":"Garside, K., Henderson, R., Makarenko, I., Masoller, C.: Topological data analysis of high resolution diabetic retinopathy images. PLoS ONE 14(5), 0217413 (2019)","journal-title":"PLoS ONE"},{"key":"200_CR27","doi-asserted-by":"crossref","DOI":"10.1016\/j.jpaa.2024.107770","volume":"228","author":"B Giunti","year":"2024","unstructured":"Giunti, B., Nolan, J.S., Otter, N., Waas, L.: Amplitudes in persistence theory. J. Pure Appl. Algebra 228, 107770 (2024)","journal-title":"J. Pure Appl. Algebra"},{"key":"200_CR28","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/0024-3795(87)90114-5","volume":"88","author":"GH Golub","year":"1987","unstructured":"Golub, G.H., Hoffman, A., Stewart, G.W.: A generalization of the Eckart-Young-Mirsky matrix approximation theorem. Linear Algebra Appl. 88, 317\u2013327 (1987)","journal-title":"Linear Algebra Appl."},{"issue":"26","key":"200_CR29","doi-asserted-by":"crossref","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":"200_CR30","unstructured":"Hofer, C., Kwitt, R., Niethammer, M., Uhl, A.: Deep learning with topological signatures. Advances Neural Inf. Process. Syst. 30 (2017)"},{"issue":"1.4","key":"200_CR31","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. ACM J. Exp. Algorithmics 22(1.4), 1\u201320 (2017)","journal-title":"ACM J. Exp. Algorithmics"},{"issue":"1","key":"200_CR32","first-page":"6947","volume":"18","author":"G Kusano","year":"2017","unstructured":"Kusano, G., Fukumizu, K., Hiraoka, Y.: Kernel method for persistence diagrams via kernel embedding and weight factor. J. Mach. Learn. Res. 18(1), 6947\u20136987 (2017)","journal-title":"J. Mach. Learn. Res."},{"issue":"3","key":"200_CR33","doi-asserted-by":"crossref","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."},{"key":"200_CR34","unstructured":"Miller, E.: Essential graded algebra over polynomial rings with real exponents. arXiv preprint arXiv:2008.03819v1 (2020)"},{"key":"200_CR35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjds\/s13688-017-0109-5","volume":"6","author":"N Otter","year":"2017","unstructured":"Otter, N., Porter, M.A., Tillmann, U., Grindrod, P., Harrington, H.A.: A roadmap for the computation of persistent homology. EPJ Data Sci. 6, 1\u201338 (2017)","journal-title":"EPJ Data Sci."},{"key":"200_CR36","unstructured":"Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. Adv. Neural Inf. Process. Syst. 32 (2019)"},{"key":"200_CR37","first-page":"2825","volume":"12","author":"F Pedregosa","year":"2011","unstructured":"Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. J. Mach. Learn. Res. 12, 2825\u20132830 (2011)","journal-title":"J. Mach. Learn. Res."},{"key":"200_CR38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.media.2019.03.014","volume":"55","author":"T Qaiser","year":"2019","unstructured":"Qaiser, T., Tsang, Y.-W., Taniyama, D., Sakamoto, N., Nakane, K., Epstein, D., Rajpoot, N.: Fast and accurate tumor segmentation of histology images using persistent homology and deep convolutional features. Med. Image Anal. 55, 1\u201314 (2019)","journal-title":"Med. Image Anal."},{"key":"200_CR39","unstructured":"Reinauer, R., Caorsi, M., Berkouk, N.: Persformer: A transformer architecture for topological machine learning. arXiv preprint arXiv:2112.15210v2 (2021)"},{"key":"200_CR40","doi-asserted-by":"crossref","unstructured":"Reininghaus, J., Huber, S., Bauer, U., Kwitt, R.: A stable multi-scale kernel for topological machine learning. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4741\u20134748 (2015)","DOI":"10.1109\/CVPR.2015.7299106"},{"key":"200_CR41","series-title":"Universitext","doi-asserted-by":"crossref","DOI":"10.1007\/b98977","volume-title":"An Introduction to Homological Algebra","author":"J Rotman","year":"2009","unstructured":"Rotman, J.: An Introduction to Homological Algebra. Universitext, 2nd edn. Springer, New York (2009)","edition":"2"},{"issue":"6","key":"200_CR42","doi-asserted-by":"crossref","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":"200_CR43","unstructured":"Skraba, P., Turner, K.: Wasserstein stability for persistence diagrams. arXiv preprint arXiv:2006.16824v5 (2020)"},{"key":"200_CR44","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511817106","volume-title":"The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities","author":"JM Steele","year":"2004","unstructured":"Steele, J.M.: The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press, Cambridge (2004)"},{"issue":"4","key":"200_CR45","doi-asserted-by":"crossref","DOI":"10.1063\/1.4978997","volume":"27","author":"BJ Stolz","year":"2017","unstructured":"Stolz, B.J., Harrington, H.A., Porter, M.A.: Persistent homology of time-dependent functional networks constructed from coupled time series. Chaos Interdiscip. J. Nonlinear Sci. 27(4), 047410 (2017)","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"issue":"9","key":"200_CR46","doi-asserted-by":"crossref","first-page":"0257215","DOI":"10.1371\/journal.pone.0257215","volume":"16","author":"R Turke\u0161","year":"2021","unstructured":"Turke\u0161, R., Nys, J., Verdonck, T., Latr\u00e9, S.: Noise robustness of persistent homology on greyscale images, across filtrations and signatures. PLoS ONE 16(9), 0257215 (2021)","journal-title":"PLoS ONE"},{"issue":"4","key":"200_CR47","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1080\/00029890.1990.11995598","volume":"97","author":"A Vince","year":"1990","unstructured":"Vince, A.: A rearrangement inequality and the permutahedron. Am. Math. Mon. 97(4), 319\u2013323 (1990)","journal-title":"Am. Math. Mon."},{"key":"200_CR48","unstructured":"Zhao, Q., Wang, Y.: Learning metrics for persistence-based summaries and applications for graph classification. Adv. Neural Inf. Process. Syst. 32 (2019)"},{"key":"200_CR49","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s00454-004-1146-y","volume":"33","author":"A Zomorodian","year":"2005","unstructured":"Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33, 249\u2013274 (2005)","journal-title":"Discrete Comput. Geom."}],"container-title":["Journal of Applied and Computational Topology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41468-024-00200-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s41468-024-00200-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41468-024-00200-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,4]],"date-time":"2025-04-04T07:30:45Z","timestamp":1743751845000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s41468-024-00200-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,20]]},"references-count":49,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,3]]}},"alternative-id":["200"],"URL":"https:\/\/doi.org\/10.1007\/s41468-024-00200-w","relation":{},"ISSN":["2367-1726","2367-1734"],"issn-type":[{"value":"2367-1726","type":"print"},{"value":"2367-1734","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,20]]},"assertion":[{"value":"4 February 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 November 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 December 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 January 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"On behalf of all authors, the corresponding author states that there is no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"4"}}