{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T09:21:08Z","timestamp":1768814468523,"version":"3.49.0"},"reference-count":49,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T00:00:00Z","timestamp":1630368000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"huawei technology co","award":["NA"],"award-info":[{"award-number":["NA"]}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/T517847\/1"],"award-info":[{"award-number":["EP\/T517847\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Entropy"],"abstract":"<jats:p>The Schr\u00f6dinger bridge problem (SBP) finds the most likely stochastic evolution between two probability distributions given a prior stochastic evolution. As well as applications in the natural sciences, problems of this kind have important applications in machine learning such as dataset alignment and hypothesis testing. Whilst the theory behind this problem is relatively mature, scalable numerical recipes to estimate the Schr\u00f6dinger bridge remain an active area of research. Our main contribution is the proof of equivalence between solving the SBP and an autoregressive maximum likelihood estimation objective. This formulation circumvents many of the challenges of density estimation and enables direct application of successful machine learning techniques. We propose a numerical procedure to estimate SBPs using Gaussian process and demonstrate the practical usage of our approach in numerical simulations and experiments.<\/jats:p>","DOI":"10.3390\/e23091134","type":"journal-article","created":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T08:42:32Z","timestamp":1630399352000},"page":"1134","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["Solving Schr\u00f6dinger Bridges via Maximum Likelihood"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2714-3357","authenticated-orcid":false,"given":"Francisco","family":"Vargas","sequence":"first","affiliation":[{"name":"The Computer Laboratory, Department of Computer Science and Technology, University of Cambridge, William Gates Building, 15 JJ Thomson Avenue, Cambridge CB3 0FD, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7791-217X","authenticated-orcid":false,"given":"Pierre","family":"Thodoroff","sequence":"additional","affiliation":[{"name":"The Computer Laboratory, Department of Computer Science and Technology, University of Cambridge, William Gates Building, 15 JJ Thomson Avenue, Cambridge CB3 0FD, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0707-0488","authenticated-orcid":false,"given":"Austen","family":"Lamacraft","sequence":"additional","affiliation":[{"name":"The Cavendish Laboratory, Deparment of Physics, The Old Schools, Trinity Ln, Cambridge CB2 1TN, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9258-1030","authenticated-orcid":false,"given":"Neil","family":"Lawrence","sequence":"additional","affiliation":[{"name":"The Computer Laboratory, Department of Computer Science and Technology, University of Cambridge, William Gates Building, 15 JJ Thomson Avenue, Cambridge CB3 0FD, UK"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,31]]},"reference":[{"key":"ref_1","unstructured":"Schr\u00f6dinger, E. (1931). Uber die Umkehrung der Naturgesetze, Akademie der Wissenschaften."},{"key":"ref_2","first-page":"269","article-title":"Sur la th\u00e9orie relativiste de l\u2019\u00e9lectron et l\u2019interpr\u00e9tation de la m\u00e9canique quantique","volume":"2","year":"1932","journal-title":"Annales de l\u2019Institut Henri Poincar\u00e9"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"343","DOI":"10.2140\/pjm.1967.21.343","article-title":"Concerning nonnegative matrices and doubly stochastic matrices","volume":"21","author":"Sinkhorn","year":"1967","journal-title":"Pac. J. Math."},{"key":"ref_4","unstructured":"Cuturi, M. (2013, January 5\u201310). Sinkhorn distances: Lightspeed computation of optimal transport. Proceedings of the 26th International Conference on Neural Information Processing Systems, Lake Tahoe, NV, USA."},{"key":"ref_5","unstructured":"Feydy, J., S\u00e9journ\u00e9, T., Vialard, F.X., Amari, S.I., Trouv\u00e9, A., and Peyr\u00e9, G. (2019, January 16). Interpolating between optimal transport and MMD using Sinkhorn divergences. Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, Okinawa, Japan."},{"key":"ref_6","unstructured":"Chizat, L., Roussillon, P., L\u00e9ger, F., Vialard, F.X., and Peyr\u00e9, G. (2020, January 6\u201312). Faster Wasserstein Distance Estimation with the Sinkhorn Divergence. Proceedings of the 2020 Advances in Neural Information Processing Systems, Vancouver, BC, Canada."},{"key":"ref_7","first-page":"20150142","article-title":"Probabilistic numerics and uncertainty in computations","volume":"471","author":"Hennig","year":"2015","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1236","DOI":"10.1214\/aoms\/1177698249","article-title":"Probability densities with given marginals","volume":"39","author":"Kullback","year":"1968","journal-title":"Ann. Math. Stat."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1160","DOI":"10.1214\/aos\/1176324703","article-title":"Convergence of the iterative proportional fitting procedure","volume":"23","author":"Ruschendorf","year":"1995","journal-title":"Ann. Stat."},{"key":"ref_10","unstructured":"Pavon, M., Tabak, E.G., and Trigila, G. (2018). The Data Driven Schr\u00f6dinger Bridge. arXiv."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Williams, C.K., and Rasmussen, C.E. (2006). Gaussian Processes for Machine Learning, MIT Press.","DOI":"10.7551\/mitpress\/3206.001.0001"},{"key":"ref_12","unstructured":"Ruttor, A., Batz, P., and Opper, M. (2013, January 5\u201310). Approximate Gaussian process inference for the drift function in stochastic differential equations. Proceedings of the 26th International Conference on Neural Information Processing Systems, Lake Tahoe, NV, USA."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"\u00d8ksendal, B. (2003). Stochastic Differential Equations, Springer.","DOI":"10.1007\/978-3-642-14394-6"},{"key":"ref_14","unstructured":"Song, Y., Sohl-Dickstein, J., Kingma, D.P., Kumar, A., Ermon, S., and Poole, B. (2020). Score-Based Generative Modeling through Stochastic Differential Equations. arXiv."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Nelson, E. (1967). Dynamical Theories of Brownian Motion, Princeton University Press.","DOI":"10.1515\/9780691219615"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/0304-4149(82)90051-5","article-title":"Reverse-time diffusion equation models","volume":"12","author":"Anderson","year":"1982","journal-title":"Stoch. Process. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/0304-4149(85)90034-1","article-title":"Reverse time diffusions","volume":"19","author":"Elliott","year":"1985","journal-title":"Stoch. Process. Appl."},{"key":"ref_18","first-page":"156","article-title":"An entropy approach to the time reversal of diffusion processes","volume":"69","author":"Follmer","year":"1984","journal-title":"Lect. Notes Control Inf. Sci."},{"key":"ref_19","unstructured":"Haussmann, U., and Pardoux, E. (1985). Time reversal of diffusion processes. Stochastic Differential Systems Filtering and Control, Springer."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Pavon, M., and Wakolbinger, A. (1991). On free energy, stochastic control, and Schr\u00f6dinger processes. Modeling, Estimation and Control of Systems with Uncertainty, Springer.","DOI":"10.1007\/978-1-4612-0443-5_22"},{"key":"ref_21","first-page":"311","article-title":"Probability measures with given marginals and conditionals: I-projections and conditional iterative proportional fitting","volume":"18","author":"Cramer","year":"2000","journal-title":"Stat. Decis.\u2014Int. J. Stoch. Methods Models"},{"key":"ref_22","unstructured":"Bernton, E., Heng, J., Doucet, A., and Jacob, P.E. (2019). Schr\u00f6dinger Bridge Samplers. arXiv."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1093\/biomet\/ass034","article-title":"Nonparametric estimation of diffusions: A differential equations approach","volume":"99","author":"Papaspiliopoulos","year":"2012","journal-title":"Biometrika"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"022109","DOI":"10.1103\/PhysRevE.98.022109","article-title":"Approximate Bayes learning of stochastic differential equations","volume":"98","author":"Batz","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1016\/j.spa.2012.08.010","article-title":"Posterior consistency via precision operators for Bayesian nonparametric drift estimation in SDEs","volume":"123","author":"Pokern","year":"2013","journal-title":"Stoch. Process. Appl."},{"key":"ref_26","unstructured":"De Bortoli, V., Thornton, J., Heng, J., and Doucet, A. (2021). Diffusion Schr\u00f6dinger Bridge with Applications to Score-Based Generative Modeling. arXiv."},{"key":"ref_27","unstructured":"Feydy, J. (2020). Geometric Data Analysis, beyond Convolutions. [Ph.D. Thesis, Universit\u00e9 Paris-Saclay]. Available online: https:\/\/www.math.ens.fr\/$\\sim$feydy."},{"key":"ref_28","unstructured":"Kingma, D.P., and Welling, M. (2013). Auto-encoding variational bayes. arXiv."},{"key":"ref_29","unstructured":"Papamakarios, G., Pavlakou, T., and Murray, I. (2017). Masked autoregressive flow for density estimation. arXiv."},{"key":"ref_30","unstructured":"Papamakarios, G. (2019). Neural density estimation and likelihood-free inference. arXiv."},{"key":"ref_31","unstructured":"Wang, G., Jiao, Y., Xu, Q., Wang, Y., and Yang, C. (2021). Deep Generative Learning via Schr\u00f6dinger Bridge. arXiv."},{"key":"ref_32","unstructured":"Huang, J., Jiao, Y., Kang, L., Liao, X., Liu, J., and Liu, Y. (2021). Schr\u00f6dinger-F\u00f6llmer Sampler: Sampling without Ergodicity. arXiv."},{"key":"ref_33","unstructured":"Kingma, D.P., Salimans, T., Poole, B., and Ho, J. (2021). Variational Diffusion Models. arXiv."},{"key":"ref_34","unstructured":"Tong, A., Huang, J., Wolf, G., Van Dijk, D., and Krishnaswamy, S. (2020, January 12\u201318). Trajectorynet: A dynamic optimal transport network for modeling cellular dynamics. Proceedings of the International Conference on Machine Learning, Online."},{"key":"ref_35","unstructured":"McCann, R.J., and Guillen, N. (2011). Five lectures on optimal transportation: Geometry, regularity and applications. Analysis and Geometry of Metric Measure Spaces: Lecture Notes of the S\u00e9minaire de Math\u00e9matiques Sup\u00e9rieure (SMS) Montr\u00e9al, American Mathematical Society."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"928","DOI":"10.1016\/j.cell.2019.01.006","article-title":"Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming","volume":"176","author":"Schiebinger","year":"2019","journal-title":"Cell"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2775","DOI":"10.1093\/bioinformatics\/btl473","article-title":"Probabilistic inference of transcription factor concentrations and gene-specific regulatory activities","volume":"22","author":"Sanguinetti","year":"2006","journal-title":"Bioinformatics"},{"key":"ref_38","unstructured":"L\u00e9onard, C. (2013). A survey of the Schr\u00f6dinger problem and some of its connections with optimal transport. arXiv."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"L\u00e9onard, C. (2014). Some properties of path measures. S\u00e9minaire de Probabilit\u00e9s XLVI, Springer.","DOI":"10.1007\/978-3-319-11970-0_8"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1080\/17442508208833209","article-title":"On backward stochastic differential equations","volume":"6","author":"Kunitha","year":"1982","journal-title":"Stochastics"},{"key":"ref_41","unstructured":"Revuz, D., and Yor, M. (2013). Continuous Martingales and Brownian Motion, Springer."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1054","DOI":"10.1214\/aoms\/1177693332","article-title":"The structure of Radon-Nikodym derivatives with respect to Wiener and related measures","volume":"42","author":"Kailath","year":"1971","journal-title":"Ann. Math. Stat."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1214\/08-BA322","article-title":"Application of Girsanov theorem to particle filtering of discretely observed continuous-time non-linear systems","volume":"3","author":"Sottinen","year":"2008","journal-title":"Bayesian Anal."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Levy, B.C. (2008). Principles of Signal Detection and Parameter Estimation, Springer.","DOI":"10.1007\/978-0-387-76544-0"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/BF01203833","article-title":"Existence of strong solutions for It\u00f4\u2019s stochastic equations via approximations","volume":"105","author":"Krylov","year":"1996","journal-title":"Probab. Theory Relat. Fields"},{"key":"ref_46","unstructured":"Sra, S. (2012, January 3\u20138). Scalable nonconvex inexact proximal splitting. Proceedings of the Advances in Neural Information Processing Systems, Lake Tahoe, NV, USA."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1561\/2200000036","article-title":"Kernels for Vector-Valued Functions: A Review","volume":"4","author":"Rosasco","year":"2012","journal-title":"Found. Trends Mach. Learn."},{"key":"ref_48","first-page":"615","article-title":"Learning multiple tasks with kernel methods","volume":"6","author":"Evgeniou","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1016\/j.sigpro.2016.08.025","article-title":"Effective sample size for importance sampling based on discrepancy measures","volume":"131","author":"Martino","year":"2017","journal-title":"Signal Process."}],"updated-by":[{"DOI":"10.3390\/e25020289","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T00:00:00Z","timestamp":1630368000000}}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1134\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,3]],"date-time":"2025-08-03T22:19:57Z","timestamp":1754259597000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1134"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,31]]},"references-count":49,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["e23091134"],"URL":"https:\/\/doi.org\/10.3390\/e23091134","relation":{"correction":[{"id-type":"doi","id":"10.3390\/e25020289","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,31]]}}}