[{"oa":1,"external_id":{"isi":["000854878500001"],"arxiv":["2102.02037"]},"day":"18","quality_controlled":"1","oa_version":"Preprint","title":"The isometry group of Wasserstein spaces: The Hilbertian case","status":"public","article_type":"original","citation":{"ama":"Gehér GP, Titkos T, Virosztek D. The isometry group of Wasserstein spaces: The Hilbertian case. <i>Journal of the London Mathematical Society</i>. 2022;106(4):3865-3894. doi:<a href=\"https://doi.org/10.1112/jlms.12676\">10.1112/jlms.12676</a>","apa":"Gehér, G. P., Titkos, T., &#38; Virosztek, D. (2022). The isometry group of Wasserstein spaces: The Hilbertian case. <i>Journal of the London Mathematical Society</i>. Wiley. <a href=\"https://doi.org/10.1112/jlms.12676\">https://doi.org/10.1112/jlms.12676</a>","ista":"Gehér GP, Titkos T, Virosztek D. 2022. The isometry group of Wasserstein spaces: The Hilbertian case. Journal of the London Mathematical Society. 106(4), 3865–3894.","chicago":"Gehér, György Pál, Tamás Titkos, and Daniel Virosztek. “The Isometry Group of Wasserstein Spaces: The Hilbertian Case.” <i>Journal of the London Mathematical Society</i>. Wiley, 2022. <a href=\"https://doi.org/10.1112/jlms.12676\">https://doi.org/10.1112/jlms.12676</a>.","short":"G.P. Gehér, T. Titkos, D. Virosztek, Journal of the London Mathematical Society 106 (2022) 3865–3894.","mla":"Gehér, György Pál, et al. “The Isometry Group of Wasserstein Spaces: The Hilbertian Case.” <i>Journal of the London Mathematical Society</i>, vol. 106, no. 4, Wiley, 2022, pp. 3865–94, doi:<a href=\"https://doi.org/10.1112/jlms.12676\">10.1112/jlms.12676</a>.","ieee":"G. P. Gehér, T. Titkos, and D. Virosztek, “The isometry group of Wasserstein spaces: The Hilbertian case,” <i>Journal of the London Mathematical Society</i>, vol. 106, no. 4. Wiley, pp. 3865–3894, 2022."},"page":"3865-3894","month":"09","year":"2022","date_updated":"2025-04-14T07:50:40Z","date_published":"2022-09-18T00:00:00Z","language":[{"iso":"eng"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","arxiv":1,"volume":106,"type":"journal_article","publication":"Journal of the London Mathematical Society","publisher":"Wiley","isi":1,"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2102.02037","open_access":"1"}],"date_created":"2023-01-16T09:46:13Z","issue":"4","author":[{"first_name":"György Pál","last_name":"Gehér","full_name":"Gehér, György Pál"},{"full_name":"Titkos, Tamás","last_name":"Titkos","first_name":"Tamás"},{"full_name":"Virosztek, Daniel","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-1109-5511","last_name":"Virosztek","first_name":"Daniel"}],"intvolume":"       106","acknowledgement":"Geher was supported by the Leverhulme Trust Early Career Fellowship (ECF-2018-125), and also by the Hungarian National Research, Development and Innovation Office - NKFIH (grant no. K115383 and K134944).\r\nTitkos was supported by the Hungarian National Research, Development and Innovation Office - NKFIH (grant no. PD128374, grant no. K115383 and K134944), by the J´anos Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the UNKP-20-5-BGE-1 New National Excellence Program of the ´Ministry of Innovation and Technology.\r\nVirosztek was supported by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 846294, by the Momentum program of the Hungarian Academy of Sciences under grant agreement no. LP2021-15/2021, and partially supported by the Hungarian National Research, Development and Innovation Office - NKFIH (grants no. K124152 and no. KH129601). ","project":[{"call_identifier":"H2020","_id":"26A455A6-B435-11E9-9278-68D0E5697425","name":"Geometric study of Wasserstein spaces and free probability","grant_number":"846294"}],"department":[{"_id":"LaEr"}],"doi":"10.1112/jlms.12676","scopus_import":"1","abstract":[{"text":"Motivated by Kloeckner’s result on the isometry group of the quadratic Wasserstein space W2(Rn), we describe the isometry group Isom(Wp(E)) for all parameters 0 < p < ∞ and for all separable real Hilbert spaces E. In particular, we show that Wp(X) is isometrically rigid for all Polish space X whenever 0 < p < 1. This is a consequence of our more general result: we prove that W1(X) is isometrically rigid if X is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters p > 1, by solving Kloeckner’s problem affirmatively on the existence of mass-splitting isometries. ","lang":"eng"}],"ec_funded":1,"keyword":["General Mathematics"],"_id":"12214","article_processing_charge":"No","publication_identifier":{"issn":["0024-6107"],"eissn":["1469-7750"]},"publication_status":"published"},{"doi":"10.1016/j.laa.2020.09.007","department":[{"_id":"LaEr"}],"scopus_import":"1","abstract":[{"lang":"eng","text":"It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence center interpretation for every symmetric Kubo-Ando mean. This characterization of the symmetric means naturally leads to a definition of weighted and multivariate versions of a large class of symmetric Kubo-Ando means. We study elementary properties of these weighted multivariate means, and note in particular that in the special case of the geometric mean we recover the weighted A#H-mean introduced by Kim, Lawson, and Lim."}],"ec_funded":1,"keyword":["Kubo-Ando mean","weighted multivariate mean","barycenter"],"publication_identifier":{"issn":["0024-3795"]},"_id":"8373","article_processing_charge":"No","publication_status":"published","main_file_link":[{"url":"https://arxiv.org/abs/2002.11678","open_access":"1"}],"date_created":"2020-09-11T08:35:50Z","author":[{"first_name":"József","last_name":"Pitrik","full_name":"Pitrik, József"},{"first_name":"Daniel","last_name":"Virosztek","orcid":"0000-0003-1109-5511","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87","full_name":"Virosztek, Daniel"}],"project":[{"grant_number":"846294","name":"Geometric study of Wasserstein spaces and free probability","call_identifier":"H2020","_id":"26A455A6-B435-11E9-9278-68D0E5697425"},{"_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme","grant_number":"291734"}],"acknowledgement":"The authors are grateful to Milán Mosonyi for fruitful discussions on the topic, and to the anonymous referee for his/her comments and suggestions.\r\nJ. Pitrik was supported by the Hungarian Academy of Sciences Lendület-Momentum Grant for Quantum Information Theory, No. 96 141, and by Hungarian National Research, Development and Innovation Office (NKFIH) via grants no. K119442, no. K124152, and no. KH129601. D. Virosztek was supported by the ISTFELLOW program of the Institute of Science and Technology Austria (project code IC1027FELL01), by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 846294, and partially supported by the Hungarian National Research, Development and Innovation Office (NKFIH) via grants no. K124152, and no. KH129601.","intvolume":"       609","year":"2021","month":"01","language":[{"iso":"eng"}],"date_published":"2021-01-15T00:00:00Z","date_updated":"2025-04-14T07:50:40Z","arxiv":1,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","volume":609,"type":"journal_article","publication":"Linear Algebra and its Applications","publisher":"Elsevier","isi":1,"day":"15","external_id":{"isi":["000581730500011"],"arxiv":["2002.11678"]},"oa":1,"oa_version":"Preprint","quality_controlled":"1","title":"A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means","status":"public","article_type":"original","citation":{"ista":"Pitrik J, Virosztek D. 2021. A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means. Linear Algebra and its Applications. 609, 203–217.","chicago":"Pitrik, József, and Daniel Virosztek. “A Divergence Center Interpretation of General Symmetric Kubo-Ando Means, and Related Weighted Multivariate Operator Means.” <i>Linear Algebra and Its Applications</i>. Elsevier, 2021. <a href=\"https://doi.org/10.1016/j.laa.2020.09.007\">https://doi.org/10.1016/j.laa.2020.09.007</a>.","ama":"Pitrik J, Virosztek D. A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means. <i>Linear Algebra and its Applications</i>. 2021;609:203-217. doi:<a href=\"https://doi.org/10.1016/j.laa.2020.09.007\">10.1016/j.laa.2020.09.007</a>","apa":"Pitrik, J., &#38; Virosztek, D. (2021). A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means. <i>Linear Algebra and Its Applications</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.laa.2020.09.007\">https://doi.org/10.1016/j.laa.2020.09.007</a>","ieee":"J. Pitrik and D. Virosztek, “A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means,” <i>Linear Algebra and its Applications</i>, vol. 609. Elsevier, pp. 203–217, 2021.","mla":"Pitrik, József, and Daniel Virosztek. “A Divergence Center Interpretation of General Symmetric Kubo-Ando Means, and Related Weighted Multivariate Operator Means.” <i>Linear Algebra and Its Applications</i>, vol. 609, Elsevier, 2021, pp. 203–17, doi:<a href=\"https://doi.org/10.1016/j.laa.2020.09.007\">10.1016/j.laa.2020.09.007</a>.","short":"J. Pitrik, D. Virosztek, Linear Algebra and Its Applications 609 (2021) 203–217."},"page":"203-217"},{"intvolume":"       380","acknowledgement":"D. Virosztek was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 846294, and partially supported by the Hungarian National Research, Development and Innovation Office (NKFIH) via grants no. K124152, and no. KH129601.","project":[{"_id":"26A455A6-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"846294","name":"Geometric study of Wasserstein spaces and free probability"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1910.10447"}],"date_created":"2021-01-22T17:55:17Z","issue":"3","author":[{"full_name":"Virosztek, Daniel","last_name":"Virosztek","orcid":"0000-0003-1109-5511","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87","first_name":"Daniel"}],"keyword":["General Mathematics"],"_id":"9036","article_processing_charge":"No","publication_identifier":{"issn":["0001-8708"]},"publication_status":"published","department":[{"_id":"LaEr"}],"doi":"10.1016/j.aim.2021.107595","scopus_import":"1","ec_funded":1,"abstract":[{"text":"In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space.","lang":"eng"}],"status":"public","article_number":"107595","article_type":"original","citation":{"ieee":"D. Virosztek, “The metric property of the quantum Jensen-Shannon divergence,” <i>Advances in Mathematics</i>, vol. 380, no. 3. Elsevier, 2021.","mla":"Virosztek, Daniel. “The Metric Property of the Quantum Jensen-Shannon Divergence.” <i>Advances in Mathematics</i>, vol. 380, no. 3, 107595, Elsevier, 2021, doi:<a href=\"https://doi.org/10.1016/j.aim.2021.107595\">10.1016/j.aim.2021.107595</a>.","short":"D. Virosztek, Advances in Mathematics 380 (2021).","ista":"Virosztek D. 2021. The metric property of the quantum Jensen-Shannon divergence. Advances in Mathematics. 380(3), 107595.","chicago":"Virosztek, Daniel. “The Metric Property of the Quantum Jensen-Shannon Divergence.” <i>Advances in Mathematics</i>. Elsevier, 2021. <a href=\"https://doi.org/10.1016/j.aim.2021.107595\">https://doi.org/10.1016/j.aim.2021.107595</a>.","ama":"Virosztek D. The metric property of the quantum Jensen-Shannon divergence. <i>Advances in Mathematics</i>. 2021;380(3). doi:<a href=\"https://doi.org/10.1016/j.aim.2021.107595\">10.1016/j.aim.2021.107595</a>","apa":"Virosztek, D. (2021). The metric property of the quantum Jensen-Shannon divergence. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2021.107595\">https://doi.org/10.1016/j.aim.2021.107595</a>"},"oa":1,"external_id":{"isi":["000619676100035"],"arxiv":["1910.10447"]},"day":"26","oa_version":"Preprint","quality_controlled":"1","title":"The metric property of the quantum Jensen-Shannon divergence","type":"journal_article","publication":"Advances in Mathematics","publisher":"Elsevier","isi":1,"month":"03","year":"2021","date_updated":"2025-04-14T07:50:40Z","language":[{"iso":"eng"}],"date_published":"2021-03-26T00:00:00Z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","arxiv":1,"volume":380},{"year":"2020","month":"08","language":[{"iso":"eng"}],"date_published":"2020-08-01T00:00:00Z","date_updated":"2025-07-10T11:54:32Z","arxiv":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":373,"type":"journal_article","publication":"Transactions of the American Mathematical Society","publisher":"American Mathematical Society","isi":1,"day":"01","external_id":{"isi":["000551418100018"],"arxiv":["2002.00859"]},"oa":1,"quality_controlled":"1","oa_version":"Preprint","title":"Isometric study of Wasserstein spaces - the real line","status":"public","article_type":"original","citation":{"apa":"Geher, G. P., Titkos, T., &#38; Virosztek, D. (2020). Isometric study of Wasserstein spaces - the real line. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/tran/8113\">https://doi.org/10.1090/tran/8113</a>","ama":"Geher GP, Titkos T, Virosztek D. Isometric study of Wasserstein spaces - the real line. <i>Transactions of the American Mathematical Society</i>. 2020;373(8):5855-5883. doi:<a href=\"https://doi.org/10.1090/tran/8113\">10.1090/tran/8113</a>","ista":"Geher GP, Titkos T, Virosztek D. 2020. Isometric study of Wasserstein spaces - the real line. Transactions of the American Mathematical Society. 373(8), 5855–5883.","chicago":"Geher, Gyorgy Pal, Tamas Titkos, and Daniel Virosztek. “Isometric Study of Wasserstein Spaces - the Real Line.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2020. <a href=\"https://doi.org/10.1090/tran/8113\">https://doi.org/10.1090/tran/8113</a>.","short":"G.P. Geher, T. Titkos, D. Virosztek, Transactions of the American Mathematical Society 373 (2020) 5855–5883.","mla":"Geher, Gyorgy Pal, et al. “Isometric Study of Wasserstein Spaces - the Real Line.” <i>Transactions of the American Mathematical Society</i>, vol. 373, no. 8, American Mathematical Society, 2020, pp. 5855–83, doi:<a href=\"https://doi.org/10.1090/tran/8113\">10.1090/tran/8113</a>.","ieee":"G. P. Geher, T. Titkos, and D. Virosztek, “Isometric study of Wasserstein spaces - the real line,” <i>Transactions of the American Mathematical Society</i>, vol. 373, no. 8. American Mathematical Society, pp. 5855–5883, 2020."},"page":"5855-5883","doi":"10.1090/tran/8113","department":[{"_id":"LaEr"}],"scopus_import":"1","abstract":[{"lang":"eng","text":"Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space W_2(R^n). It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute Isom(W_p(R)), the isometry group of the Wasserstein space\r\nW_p(R) for all p \\in [1,\\infty) \\setminus {2}. We show that W_2(R) is also exceptional regarding the\r\nparameter p: W_p(R) is isometrically rigid if and only if p is not equal to 2. Regarding the underlying\r\nspace, we prove that the exceptionality of p = 2 disappears if we replace R by the compact\r\ninterval [0,1]. Surprisingly, in that case, W_p([0,1]) is isometrically rigid if and only if\r\np is not equal to 1. Moreover, W_1([0,1]) admits isometries that split mass, and Isom(W_1([0,1]))\r\ncannot be embedded into Isom(W_1(R))."}],"ec_funded":1,"keyword":["Wasserstein space","isometric embeddings","isometric rigidity","exotic isometry flow"],"publication_identifier":{"eissn":["1088-6850"],"issn":["0002-9947"]},"_id":"7389","article_processing_charge":"No","publication_status":"published","main_file_link":[{"url":"https://arxiv.org/abs/2002.00859","open_access":"1"}],"ddc":["515"],"issue":"8","date_created":"2020-01-29T10:20:46Z","author":[{"first_name":"Gyorgy Pal","last_name":"Geher","full_name":"Geher, Gyorgy Pal"},{"last_name":"Titkos","first_name":"Tamas","full_name":"Titkos, Tamas"},{"orcid":"0000-0003-1109-5511","last_name":"Virosztek","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87","first_name":"Daniel","full_name":"Virosztek, Daniel"}],"project":[{"call_identifier":"H2020","_id":"26A455A6-B435-11E9-9278-68D0E5697425","grant_number":"846294","name":"Geometric study of Wasserstein spaces and free probability"}],"intvolume":"       373"},{"isi":1,"publisher":"Springer Nature","publication":"Letters in Mathematical Physics","type":"journal_article","volume":110,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","arxiv":1,"date_updated":"2025-10-09T08:23:15Z","language":[{"iso":"eng"}],"date_published":"2020-08-01T00:00:00Z","month":"08","year":"2020","page":"2039-2052","article_type":"original","citation":{"mla":"Pitrik, Jozsef, and Daniel Virosztek. “Quantum Hellinger Distances Revisited.” <i>Letters in Mathematical Physics</i>, vol. 110, no. 8, Springer Nature, 2020, pp. 2039–52, doi:<a href=\"https://doi.org/10.1007/s11005-020-01282-0\">10.1007/s11005-020-01282-0</a>.","short":"J. Pitrik, D. Virosztek, Letters in Mathematical Physics 110 (2020) 2039–2052.","ieee":"J. Pitrik and D. Virosztek, “Quantum Hellinger distances revisited,” <i>Letters in Mathematical Physics</i>, vol. 110, no. 8. Springer Nature, pp. 2039–2052, 2020.","apa":"Pitrik, J., &#38; Virosztek, D. (2020). Quantum Hellinger distances revisited. <i>Letters in Mathematical Physics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11005-020-01282-0\">https://doi.org/10.1007/s11005-020-01282-0</a>","ama":"Pitrik J, Virosztek D. Quantum Hellinger distances revisited. <i>Letters in Mathematical Physics</i>. 2020;110(8):2039-2052. doi:<a href=\"https://doi.org/10.1007/s11005-020-01282-0\">10.1007/s11005-020-01282-0</a>","chicago":"Pitrik, Jozsef, and Daniel Virosztek. “Quantum Hellinger Distances Revisited.” <i>Letters in Mathematical Physics</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s11005-020-01282-0\">https://doi.org/10.1007/s11005-020-01282-0</a>.","ista":"Pitrik J, Virosztek D. 2020. Quantum Hellinger distances revisited. Letters in Mathematical Physics. 110(8), 2039–2052."},"status":"public","title":"Quantum Hellinger distances revisited","corr_author":"1","oa_version":"Preprint","quality_controlled":"1","oa":1,"day":"01","external_id":{"isi":["000551556000002"],"arxiv":["1903.10455"]},"publication_status":"published","_id":"7618","article_processing_charge":"No","publication_identifier":{"eissn":["1573-0530"],"issn":["0377-9017"]},"ec_funded":1,"abstract":[{"lang":"eng","text":"This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. (Lett Math Phys 109:1777–1804, 2019) with a particular emphasis on barycenters. We introduce the family of generalized quantum Hellinger divergences that are of the form ϕ(A,B)=Tr((1−c)A+cB−AσB), where σ is an arbitrary Kubo–Ando mean, and c∈(0,1) is the weight of σ. We note that these divergences belong to the family of maximal quantum f-divergences, and hence are jointly convex, and satisfy the data processing inequality. We derive a characterization of the barycenter of finitely many positive definite operators for these generalized quantum Hellinger divergences. We note that the characterization of the barycenter as the weighted multivariate 1/2-power mean, that was claimed in Bhatia et al. (2019), is true in the case of commuting operators, but it is not correct in the general case. "}],"scopus_import":"1","department":[{"_id":"LaEr"}],"doi":"10.1007/s11005-020-01282-0","acknowledgement":"J. Pitrik was supported by the Hungarian Academy of Sciences Lendület-Momentum Grant for Quantum\r\nInformation Theory, No. 96 141, and by the Hungarian National Research, Development and Innovation\r\nOffice (NKFIH) via Grants Nos. K119442, K124152 and KH129601. D. Virosztek was supported by the\r\nISTFELLOW program of the Institute of Science and Technology Austria (Project Code IC1027FELL01),\r\nby the European Union’s Horizon 2020 research and innovation program under the Marie\r\nSklodowska-Curie Grant Agreement No. 846294, and partially supported by the Hungarian National\r\nResearch, Development and Innovation Office (NKFIH) via Grants Nos. K124152 and KH129601.\r\nWe are grateful to Milán Mosonyi for drawing our attention to Ref.’s [6,14,15,17,\r\n20,21], for comments on earlier versions of this paper, and for several discussions on the topic. We are\r\nalso grateful to Miklós Pálfia for several discussions; to László Erdös for his essential suggestions on the\r\nstructure and highlights of this paper, and for his comments on earlier versions; and to the anonymous\r\nreferee for his/her valuable comments and suggestions.","intvolume":"       110","project":[{"call_identifier":"H2020","_id":"26A455A6-B435-11E9-9278-68D0E5697425","grant_number":"846294","name":"Geometric study of Wasserstein spaces and free probability"},{"name":"International IST Postdoc Fellowship Programme","grant_number":"291734","call_identifier":"FP7","_id":"25681D80-B435-11E9-9278-68D0E5697425"}],"author":[{"full_name":"Pitrik, Jozsef","last_name":"Pitrik","first_name":"Jozsef"},{"full_name":"Virosztek, Daniel","first_name":"Daniel","last_name":"Virosztek","orcid":"0000-0003-1109-5511","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87"}],"date_created":"2020-03-25T15:57:48Z","issue":"8","main_file_link":[{"url":"https://arxiv.org/abs/1903.10455","open_access":"1"}]}]