{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:30:37Z","timestamp":1772119837538,"version":"3.50.1"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,9,2]],"date-time":"2025-09-02T00:00:00Z","timestamp":1756771200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,9,2]],"date-time":"2025-09-02T00:00:00Z","timestamp":1756771200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001824","name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky","doi-asserted-by":"publisher","award":["P103\/22-11101S"],"award-info":[{"award-number":["P103\/22-11101S"]}],"id":[{"id":"10.13039\/501100001824","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001823","name":"Ministerstvo \u0160kolstv\u00ed, Ml\u00e1de\u017ee a T\u011blov\u00fdchovy","doi-asserted-by":"publisher","award":["CZ.02.01.01\/00\/22 008\/0004590"],"award-info":[{"award-number":["CZ.02.01.01\/00\/22 008\/0004590"]}],"id":[{"id":"10.13039\/501100001823","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2025,12]]},"DOI":"10.1007\/s11222-025-10707-6","type":"journal-article","created":{"date-parts":[[2025,9,2]],"date-time":"2025-09-02T06:14:59Z","timestamp":1756793699000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Tensor train approximation of multivariate Gaussian density by scaling and squaring"],"prefix":"10.1007","volume":"35","author":[{"given":"Ji\u0159\u00ed","family":"Ajgl","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ond\u0159ej","family":"Straka","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,9,2]]},"reference":[{"key":"10707_CR1","doi-asserted-by":"publisher","unstructured":"Ajgl, J., Straka, O.: On fusion of probability density functions using tensor train decomposition. In: 27th International Conference on Information Fusion, Venice, Italy (2024). https:\/\/doi.org\/10.23919\/FUSION59988.2024.10706475","DOI":"10.23919\/FUSION59988.2024.10706475"},{"key":"10707_CR2","doi-asserted-by":"publisher","unstructured":"Ajgl, J., Straka, O.: Aspects of density approximation by tensor trains. In: 28th International Conference on Information Fusion, Rio de Janeiro, Brazil (2025). https:\/\/doi.org\/10.48550\/arXiv.2505.22218","DOI":"10.48550\/arXiv.2505.22218"},{"key":"10707_CR3","doi-asserted-by":"publisher","unstructured":"Bhattarai, M., Chennupati, G., Skau, E., Vangara, R., Djidjev, H., Alexandrov, B.S.: Distributed non-negative tensor train decomposition. In: 2020 IEEE High Performance Extreme Computing Conference (HPEC), Waltham, Massachusetts, USA (2020). https:\/\/doi.org\/10.1109\/HPEC43674.2020.9286234","DOI":"10.1109\/HPEC43674.2020.9286234"},{"issue":"4","key":"10707_CR4","doi-asserted-by":"publisher","first-page":"2405","DOI":"10.1137\/15M1036919","volume":"38","author":"D Bigoni","year":"2016","unstructured":"Bigoni, D., Engsig-Karup, A.P., Marzouk, Y.M.: Spectral tensor-train decomposition. SIAM J. Sci. Comput. 38(4), 2405\u20132439 (2016). https:\/\/doi.org\/10.1137\/15M1036919","journal-title":"SIAM J. Sci. Comput."},{"key":"10707_CR5","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1007\/s11222-019-09910-z","volume":"30","author":"SV Dolgov","year":"2020","unstructured":"Dolgov, S.V., Anaya-Izquierdo, K., Fox, C., Scheichl, R.: Approximation and sampling of multivariate probability distributions in the tensor train decomposition. Stat. Comput. 30, 603\u2013625 (2020). https:\/\/doi.org\/10.1007\/s11222-019-09910-z","journal-title":"Stat. Comput."},{"issue":"5","key":"10707_CR6","doi-asserted-by":"publisher","first-page":"2248","DOI":"10.1137\/140953289","volume":"36","author":"SV Dolgov","year":"2014","unstructured":"Dolgov, S.V., Savostyanov, D.V.: Alternating minimal energy methods for linear systems in higher dimensions. SIAM J. Sci. Comput. 36(5), 2248\u20132271 (2014). https:\/\/doi.org\/10.1137\/140953289","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"10707_CR7","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1109\/TAES.2018.2850210","volume":"55","author":"J Dun\u00edk","year":"2019","unstructured":"Dun\u00edk, J., Sot\u00e1k, M., Vesel\u00fd, M., Straka, O., Hawkinson, W.: Design of Rao-Blackwellized point-mass filter with application in terrain aided navigation. IEEE Trans. Aerosp. Electron. Syst. 55(1), 251\u2013272 (2019). https:\/\/doi.org\/10.1109\/TAES.2018.2850210","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"10707_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmva.2020.104619","volume":"178","author":"PJ Forrester","year":"2020","unstructured":"Forrester, P.J., Zhang, J.: Parametrising correlation matrices. J. Multivar. Anal. 178, 104619 (2020). https:\/\/doi.org\/10.1016\/j.jmva.2020.104619","journal-title":"J. Multivar. Anal."},{"key":"10707_CR9","unstructured":"Govaers, F., Demissie, B., Khan, A., Ulmke, M., Koch, W.: Tensor decomposition-based multitarget tracking in cluttered environments. Journal of advances in information fusion 14(1), 86\u201397 (2019)"},{"issue":"15","key":"10707_CR10","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1016\/j.cma.2018.12.015","volume":"347","author":"A Gorodetsky","year":"2019","unstructured":"Gorodetsky, A., Karaman, S., Marzouk, Y.: A continuous analogue of the tensor-train decomposition. Comput. Methods Appl. Mech. Eng. 347(15), 59\u201384 (2019). https:\/\/doi.org\/10.1016\/j.cma.2018.12.015","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"1","key":"10707_CR11","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1002\/gamm.201310004","volume":"36","author":"L Grasedyck","year":"2013","unstructured":"Grasedyck, L., Kressner, D., Tobler, C.: A literature survey of low-rank tensor approximation techniques. GAMM-Mitteilungen 36(1), 53\u201378 (2013). https:\/\/doi.org\/10.1002\/gamm.201310004","journal-title":"GAMM-Mitteilungen"},{"key":"10707_CR12","doi-asserted-by":"publisher","unstructured":"Govaers, F.: On a CPD decomposition of a multi-variate Gaussian. In: 2018 Sensor Data Fusion: Trends, Solutions, Applications (SDF), Bonn, Germany (2018). https:\/\/doi.org\/10.1109\/SDF.2018.8547115","DOI":"10.1109\/SDF.2018.8547115"},{"key":"10707_CR13","doi-asserted-by":"publisher","unstructured":"Govaers, F.: On canonical polyadic decomposition of non-linear Gaussian likelihood functions. In: 21st International Conference on Information Fusion, Cambridge, UK, pp. 1107\u20131113 (2018). https:\/\/doi.org\/10.23919\/ICIF.2018.8455702","DOI":"10.23919\/ICIF.2018.8455702"},{"issue":"1\u20133","key":"10707_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0024-3795(96)00301-1","volume":"261","author":"SA Goreinov","year":"1997","unstructured":"Goreinov, S.A., Tyrtyshnikov, E.E., Zamarashkin, N.L.: A theory of pseudoskeleton approximations. Linear Algebra Appl. 261(1\u20133), 1\u201321 (1997). https:\/\/doi.org\/10.1016\/S0024-3795(96)00301-1","journal-title":"Linear Algebra Appl."},{"key":"10707_CR15","doi-asserted-by":"publisher","DOI":"10.56021\/9781421407944","volume-title":"Matrix Computations","author":"GH Golub","year":"2013","unstructured":"Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)","edition":"4"},{"key":"10707_CR16","volume-title":"Matrix Analysis","author":"RA Horn","year":"2013","unstructured":"Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)","edition":"2"},{"issue":"1","key":"10707_CR17","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1109\/TSP.2013.2285514","volume":"62","author":"K Huang","year":"2014","unstructured":"Huang, K., Sidiropoulos, N.D., Swami, A.: Non-negative matrix factorization revisited: Uniqueness and algorithm for symmetric decomposition. IEEE Trans. Signal Process. 62(1), 211\u2013224 (2014). https:\/\/doi.org\/10.1109\/TSP.2013.2285514","journal-title":"IEEE Trans. Signal Process."},{"key":"10707_CR18","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1007\/s00365-011-9131-1","volume":"34","author":"BN Khoromskij","year":"2011","unstructured":"Khoromskij, B.N.: O($$d$$ log $${N}$$)-quantics approximation of $${N}$$-$$d$$ tensors in high-dimensional numerical modeling. Constr. Approx. 34, 257\u2013280 (2011). https:\/\/doi.org\/10.1007\/s00365-011-9131-1","journal-title":"Constr. Approx."},{"issue":"3","key":"10707_CR19","doi-asserted-by":"publisher","first-page":"2467","DOI":"10.1002\/nla.2467","volume":"30","author":"A Litvinenko","year":"2023","unstructured":"Litvinenko, A., Marzouk, Y., Matthies, H.G., Scavino, M., Spantini, A.: Computing $$f$$-divergences and distances of high-dimensional probability density functions. Numerical Linear Algebra with Applications 30(3), 2467 (2023). https:\/\/doi.org\/10.1002\/nla.2467","journal-title":"Numerical Linear Algebra with Applications"},{"key":"10707_CR20","doi-asserted-by":"publisher","unstructured":"Matou\u0161ek, J., Brandner, M., Dun\u00edk, J., Pun\u010doch\u00e1\u0159, I.: Tensor train discrete grid-based filters: Breaking the curse of dimensionality. In: 20th IFAC Symposium on System Identification, Boston, Massachusetts, USA, pp. 19\u201324 (2024). https:\/\/doi.org\/10.1016\/j.ifacol.2024.08.498","DOI":"10.1016\/j.ifacol.2024.08.498"},{"issue":"5","key":"10707_CR21","doi-asserted-by":"publisher","first-page":"2295","DOI":"10.1137\/090752286","volume":"33","author":"I Oseledets","year":"2011","unstructured":"Oseledets, I.: Tensor-train decomposition. SIAM Journal of Scientific Computing 33(5), 2295\u20132317 (2011). https:\/\/doi.org\/10.1137\/090752286","journal-title":"SIAM Journal of Scientific Computing"},{"key":"10707_CR22","unstructured":"Oseledets, I.V.: TT-Toolbox 2.3. (2014). https:\/\/github.com\/oseledets\/TT-Toolbox"},{"issue":"3","key":"10707_CR23","doi-asserted-by":"publisher","first-page":"1191","DOI":"10.1137\/20M1314653","volume":"10","author":"PB Rohrbach","year":"2022","unstructured":"Rohrbach, P.B., Dolgov, S., Grasedyck, L., Scheichl, R.: Rank bounds for approximating gaussian densities in the tensor-train format. SIAM\/ASA Journal on Uncertainty Quantification 10(3), 1191\u20131224 (2022). https:\/\/doi.org\/10.1137\/20M1314653","journal-title":"SIAM\/ASA Journal on Uncertainty Quantification"},{"issue":"4","key":"10707_CR24","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1080\/00031305.1994.10476079","volume":"48","author":"PJ Rousseeuw","year":"1994","unstructured":"Rousseeuw, P.J., Molenberghs, G.: The shape of correlation matrices. Am. Stat. 48(4), 276\u2013279 (1994). https:\/\/doi.org\/10.1080\/00031305.1994.10476079","journal-title":"Am. Stat."},{"issue":"1","key":"10707_CR25","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1080\/00031305.1988.10475524","volume":"42","author":"JL Rodgers","year":"1988","unstructured":"Rodgers, J.L., Nicewander, W.A.: Thirteen ways to look at the correlation coefficient. Am. Stat. 42(1), 59\u201366 (1988). https:\/\/doi.org\/10.1080\/00031305.1988.10475524","journal-title":"Am. Stat."},{"key":"10707_CR26","doi-asserted-by":"publisher","unstructured":"Rabanser, S., Shchur, O., G\u00fcnnemann, S.: Introduction to tensor decompositions and their applications in machine learning. ArXiv (2017) https:\/\/doi.org\/10.48550\/arXiv.1711.10781","DOI":"10.48550\/arXiv.1711.10781"},{"key":"10707_CR27","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1016\/j.laa.2014.06.006","volume":"458","author":"DV Savostyanov","year":"2014","unstructured":"Savostyanov, D.V.: Quasioptimality of maximum-volume cross interpolation of tensors. Linear Algebra Appl. 458, 217\u2013244 (2014). https:\/\/doi.org\/10.1016\/j.laa.2014.06.006","journal-title":"Linear Algebra Appl."},{"key":"10707_CR28","unstructured":"Sun, Y., Kumar, M.: Nonlinear Bayesian filtering based on Fokker-Planck equation and tensor decomposition. In: 18th International Conference on Information Fusion, Washington, DC, USA (2015). https:\/\/ieeexplore.ieee.org\/document\/7266732"},{"issue":"7","key":"10707_CR29","doi-asserted-by":"publisher","first-page":"1135","DOI":"10.1016\/j.automatica.2006.03.010","volume":"42","author":"M \u0160imandl","year":"2006","unstructured":"\u0160imandl, M., Kr\u00e1lovec, J., S\u00f6derstr\u00f6m, T.: Advanced point-mass method for nonlinear state estimation. Automatica 42(7), 1135\u20131145 (2006). https:\/\/doi.org\/10.1016\/j.automatica.2006.03.010","journal-title":"Automatica"},{"key":"10707_CR30","doi-asserted-by":"publisher","unstructured":"Savostyanov, D., Oseledets, I.: Fast adaptive interpolation of multi-dimensional arrays in tensor train format. In: The 2011 International Workshop on Multidimensional (nD) Systems, Poitiers, France (2011). https:\/\/doi.org\/10.1109\/nDS.2011.6076873","DOI":"10.1109\/nDS.2011.6076873"},{"key":"10707_CR31","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1109\/TSP.2023.3240359","volume":"71","author":"P Tichavsk\u00fd","year":"2023","unstructured":"Tichavsk\u00fd, P., Straka, O., Dun\u00edk, J.: Grid-based bayesian filters with functional decomposition of transient density. IEEE Trans. Signal Process. 71, 92\u2013104 (2023). https:\/\/doi.org\/10.1109\/TSP.2023.3240359","journal-title":"IEEE Trans. Signal Process."},{"issue":"2173","key":"10707_CR32","doi-asserted-by":"publisher","first-page":"20140585","DOI":"10.1098\/rspa.2014.0585","volume":"471","author":"A Townsend","year":"2015","unstructured":"Townsend, A., Trefethen, L.N.: Continuous analogues of matrix factorizations. Proceedings of the Royal Society A 471(2173), 20140585 (2015). https:\/\/doi.org\/10.1098\/rspa.2014.0585","journal-title":"Proceedings of the Royal Society A"},{"key":"10707_CR33","doi-asserted-by":"publisher","unstructured":"Zhao, Y., Cui, T.: Tensor-train methods for sequential state and parameter learning in state-space models. ArXiv (2023). https:\/\/doi.org\/10.48550\/arXiv.2301.09891","DOI":"10.48550\/arXiv.2301.09891"},{"issue":"12","key":"10707_CR34","doi-asserted-by":"publisher","first-page":"4990","DOI":"10.1109\/TIP.2015.2478396","volume":"24","author":"G Zhou","year":"2015","unstructured":"Zhou, G., Cichocki, A., Zhao, Q., Xie, S.: Efficient nonnegative Tucker decompositions: Algorithms and uniqueness. IEEE Trans. Image Process. 24(12), 4990\u20135003 (2015). https:\/\/doi.org\/10.1109\/TIP.2015.2478396","journal-title":"IEEE Trans. Image Process."}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-025-10707-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-025-10707-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-025-10707-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T07:17:10Z","timestamp":1762327030000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-025-10707-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,2]]},"references-count":34,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["10707"],"URL":"https:\/\/doi.org\/10.1007\/s11222-025-10707-6","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-5895329\/v1","asserted-by":"object"}]},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,2]]},"assertion":[{"value":"24 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 August 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 September 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"182"}}