{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T04:53:35Z","timestamp":1769748815310,"version":"3.49.0"},"reference-count":33,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,17]],"date-time":"2022-08-17T00:00:00Z","timestamp":1660694400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100008398","name":"Villum Foundation","doi-asserted-by":"publisher","award":["22924"],"award-info":[{"award-number":["22924"]}],"id":[{"id":"10.13039\/100008398","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100008398","name":"Villum Foundation","doi-asserted-by":"publisher","award":["40582"],"award-info":[{"award-number":["40582"]}],"id":[{"id":"10.13039\/100008398","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100008398","name":"Villum Foundation","doi-asserted-by":"publisher","award":["NNF18OC0052000"],"award-info":[{"award-number":["NNF18OC0052000"]}],"id":[{"id":"10.13039\/100008398","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100008398","name":"Villum Foundation","doi-asserted-by":"publisher","award":["DMS-1912030"],"award-info":[{"award-number":["DMS-1912030"]}],"id":[{"id":"10.13039\/100008398","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Novo Nordisk Foundation","award":["22924"],"award-info":[{"award-number":["22924"]}]},{"name":"Novo Nordisk Foundation","award":["40582"],"award-info":[{"award-number":["40582"]}]},{"name":"Novo Nordisk Foundation","award":["NNF18OC0052000"],"award-info":[{"award-number":["NNF18OC0052000"]}]},{"name":"Novo Nordisk Foundation","award":["DMS-1912030"],"award-info":[{"award-number":["DMS-1912030"]}]},{"name":"National Science Foundation","award":["22924"],"award-info":[{"award-number":["22924"]}]},{"name":"National Science Foundation","award":["40582"],"award-info":[{"award-number":["40582"]}]},{"name":"National Science Foundation","award":["NNF18OC0052000"],"award-info":[{"award-number":["NNF18OC0052000"]}]},{"name":"National Science Foundation","award":["DMS-1912030"],"award-info":[{"award-number":["DMS-1912030"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>We present schemes for simulating Brownian bridges on complete and connected Lie groups and homogeneous spaces. We use this to construct an estimation scheme for recovering an unknown left- or right-invariant Riemannian metric on the Lie group from samples. We subsequently show how pushing forward the distributions generated by Brownian motions on the group results in distributions on homogeneous spaces that exhibit a non-trivial covariance structure. The pushforward measure gives rise to new non-parametric families of distributions on commonly occurring spaces such as spheres and symmetric positive tensors. We extend the estimation scheme to fit these distributions to homogeneous space-valued data. We demonstrate both the simulation schemes and estimation procedures on Lie groups and homogenous spaces, including SPD(3)=GL+(3)\/SO(3) and S2=SO(3)\/SO(2).<\/jats:p>","DOI":"10.3390\/a15080290","type":"journal-article","created":{"date-parts":[[2022,8,17]],"date-time":"2022-08-17T03:15:27Z","timestamp":1660706127000},"page":"290","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Discrete-Time Observations of Brownian Motion on Lie Groups and Homogeneous Spaces: Sampling and Metric Estimation"],"prefix":"10.3390","volume":"15","author":[{"given":"Mathias H\u00f8jgaard","family":"Jensen","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Copenhagen, 2100 Copenhagen, Denmark"}]},{"given":"Sarang","family":"Joshi","sequence":"additional","affiliation":[{"name":"Department of Biomedical Engineering, University of Utah, Salt Lake City, UT 84112, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6784-0328","authenticated-orcid":false,"given":"Stefan","family":"Sommer","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Copenhagen, 2100 Copenhagen, Denmark"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"257","DOI":"10.2307\/3318480","article-title":"Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes","volume":"1","author":"Pedersen","year":"1995","journal-title":"Bernoulli"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"645","DOI":"10.3150\/12-BEJ501","article-title":"Simple simulation of diffusion bridges with application to likelihood inference for diffusions","volume":"20","author":"Bladt","year":"2014","journal-title":"Bernoulli"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1111\/rssb.12118","article-title":"Simulation of multivariate diffusion bridges","volume":"78","author":"Bladt","year":"2016","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_4","unstructured":"Bui, M.N., Pokern, Y., and Dellaportas, P. (2021). Inference for partially observed Riemannian Ornstein\u2013Uhlenbeck diffusions of covariance matrices. arXiv."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1660","DOI":"10.1016\/j.spa.2006.04.004","article-title":"Simulation of Conditioned Diffusion and Application to Parameter Estimation","volume":"116","author":"Delyon","year":"2006","journal-title":"Stoch. Process. Their Appl."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Jensen, M.H., Mallasto, A., and Sommer, S. (2019, January 27\u201329). Simulation of Conditioned Diffusions on the Flat Torus. Proceedings of the International Conference on Geometric Science of Information, Toulouse, France.","DOI":"10.1007\/978-3-030-26980-7_71"},{"key":"ref_7","unstructured":"Jensen, M.H., and Sommer, S. (2021). Simulation of Conditioned Semimartingales on Riemannian Manifolds. arXiv."},{"key":"ref_8","first-page":"2358","article-title":"Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals","volume":"11","author":"Schauer","year":"2017","journal-title":"Electron. J. Stat."},{"key":"ref_9","unstructured":"Papaspiliopoulos, O., and Roberts, G. (2012). Importance sampling techniques for estimation of diffusion models. Stat. Methods Stoch. Differ. Equ., 311\u2013340."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2917","DOI":"10.3150\/16-BEJ833","article-title":"Guided proposals for simulating multi-dimensional diffusion bridges","volume":"23","author":"Schauer","year":"2017","journal-title":"Bernoulli"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Sommer, S., Arnaudon, A., Kuhnel, L., and Joshi, S. (2017, January 10\u201314). Bridge Simulation and Metric Estimation on Landmark Manifolds. Proceedings of the Graphs in Biomedical Image Analysis, Computational Anatomy and Imaging Genetics, Lecture Notes in Computer Science, Quebec, QC, Canada.","DOI":"10.1007\/978-3-319-67675-3_8"},{"key":"ref_12","unstructured":"Bui, M.N. (2022). Inference on Riemannian Manifolds: Regression and Stochastic Differential Equations. [Ph.D. Thesis, UCL (University College London)]."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1098\/rspa.1953.0064","article-title":"Dispersion on a Sphere","volume":"217","author":"Fisher","year":"1953","journal-title":"Proc. R. Soc. Lond. A Math. Phys. Eng. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1111\/j.2517-6161.1982.tb01189.x","article-title":"The Fisher-Bingham Distribution on the Sphere","volume":"44","author":"Kent","year":"1982","journal-title":"J. R. Stat. Soc. Ser. B (Methodol.)"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1007\/s11118-017-9667-1","article-title":"Brownian bridges to submanifolds","volume":"49","author":"Thompson","year":"2018","journal-title":"Potential Anal."},{"key":"ref_16","first-page":"1","article-title":"Langevin Diffusions on the Torus: Estimation and Applications","volume":"29","author":"Mardia","year":"2017","journal-title":"Stat. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Hamelryck, T., Kent, J.T., and Krogh, A. (2006). Sampling Realistic Protein Conformations Using Local Structural Bias. PLoS Comput. Biol., 2.","DOI":"10.1371\/journal.pcbi.0020131"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/s11263-005-3222-z","article-title":"A Riemannian Framework for Tensor Computing","volume":"66","author":"Pennec","year":"2006","journal-title":"Int. J. Comput. Vis."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"S161","DOI":"10.1016\/j.neuroimage.2004.07.023","article-title":"Statistics on Diffeomorphisms via Tangent Space Representations","volume":"23","author":"Vaillant","year":"2004","journal-title":"NeuroImage"},{"key":"ref_20","unstructured":"Yang, L. (2011). Means of Probability Measures in Riemannian Manifolds and Applications to Radar Target Detection. [Ph.D. Thesis, Poitiers University]."},{"key":"ref_21","unstructured":"Grenander, U. (1963). Probabilities on Algebraic Structures, Wiley."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Nielsen, F., and Barbaresco, F. (2021). Bridge Simulation and Metric Estimation on Lie Groups. Proceedings of the Geometric Science of Information, Springer International Publishing. Lecture Notes in Computer Science.","DOI":"10.1007\/978-3-030-80209-7"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Pennec, X., Sommer, S., and Fletcher, T. (2020). Riemannian Geometric Statistics in Medical Image Analysis, Elsevier.","DOI":"10.1016\/B978-0-12-814725-2.00012-1"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Liao, M. (2004). L\u00e9vy Processes in Lie Groups, Cambridge University Press.","DOI":"10.1017\/CBO9780511546624"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Shigekawa, I. (1984). Transformations of the Brownian motion on a Riemannian symmetric space. Zeitschrift f\u00fcr Wahrscheinlichkeitstheorie und Verwandte Gebiete.","DOI":"10.1007\/BF00531836"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1491","DOI":"10.1214\/aop\/1176991988","article-title":"The radial part of Brownian motion on a manifold: A semimartingale property","volume":"15","author":"Kendall","year":"1987","journal-title":"Ann. Probab."},{"key":"ref_27","first-page":"369","article-title":"Some consequences of the nature of the distance function on the cut locus in a Riemannian manifold","volume":"56","author":"Barden","year":"1997","journal-title":"J. LMS"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/BF01192198","article-title":"It\u00f4 correction terms for the radial parts of semimartingales on manifolds","volume":"101","author":"Le","year":"1995","journal-title":"Probab. Theory Relat. Fields"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Hsu, E.P. (2002). Stochastic Analysis on Manifolds, AMS.","DOI":"10.1090\/gsm\/038"},{"key":"ref_30","unstructured":"Thompson, J. (2015). Submanifold Bridge Processes. [Ph.D. Thesis, University of Warwick]."},{"key":"ref_31","unstructured":"Hansen, P., Eltzner, B., Huckemann, S.F., and Sommer, S. (2021). Diffusion Means in Geometric Spaces. arXiv."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Hansen, P., Eltzner, B., and Sommer, S. (2021). Diffusion Means and Heat Kernel on Manifolds. arXiv.","DOI":"10.1007\/978-3-030-80209-7_13"},{"key":"ref_33","first-page":"411","article-title":"Differential Geometry and Stochastic Dynamics with Deep Learning Numerics","volume":"356","author":"Sommer","year":"2019","journal-title":"Appl. Math. Comput."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/8\/290\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:10:51Z","timestamp":1760141451000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/8\/290"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,17]]},"references-count":33,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["a15080290"],"URL":"https:\/\/doi.org\/10.3390\/a15080290","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,8,17]]}}}