{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T06:44:35Z","timestamp":1762325075289,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:00:00Z","timestamp":1627603200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002241","name":"Japan Science and Technology Agency","doi-asserted-by":"publisher","award":["JPMJAX190R"],"award-info":[{"award-number":["JPMJAX190R"]}],"id":[{"id":"10.13039\/501100002241","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Langevin dynamics (LD) has been extensively studied theoretically and practically as a basic sampling technique. Recently, the incorporation of non-reversible dynamics into LD is attracting attention because it accelerates the mixing speed of LD. Popular choices for non-reversible dynamics include underdamped Langevin dynamics (ULD), which uses second-order dynamics and perturbations with skew-symmetric matrices. Although ULD has been widely used in practice, the application of skew acceleration is limited although it is expected to show superior performance theoretically. Current work lacks a theoretical understanding of issues that are important to practitioners, including the selection criteria for skew-symmetric matrices, quantitative evaluations of acceleration, and the large memory cost of storing skew matrices. In this study, we theoretically and numerically clarify these problems by analyzing acceleration focusing on how the skew-symmetric matrix perturbs the Hessian matrix of potential functions. We also present a practical algorithm that accelerates the standard LD and ULD, which uses novel memory-efficient skew-symmetric matrices under parallel-chain Monte Carlo settings.<\/jats:p>","DOI":"10.3390\/e23080993","type":"journal-article","created":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T12:59:24Z","timestamp":1627649964000},"page":"993","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Accelerated Diffusion-Based Sampling by the Non-Reversible Dynamics with Skew-Symmetric Matrices"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0661-0729","authenticated-orcid":false,"given":"Futoshi","family":"Futami","sequence":"first","affiliation":[{"name":"Communication Science Laboratories, NTT, Hikaridai, Seika-cho, \u201cKeihanna Science City\u201d, Kyoto 619-0237, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tomoharu","family":"Iwata","sequence":"additional","affiliation":[{"name":"Communication Science Laboratories, NTT, Hikaridai, Seika-cho, \u201cKeihanna Science City\u201d, Kyoto 619-0237, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Naonori","family":"Ueda","sequence":"additional","affiliation":[{"name":"Communication Science Laboratories, NTT, Hikaridai, Seika-cho, \u201cKeihanna Science City\u201d, Kyoto 619-0237, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Issei","family":"Sato","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,30]]},"reference":[{"key":"ref_1","unstructured":"Murphy, K.P. (2012). Machine Learning: A Probabilistic Perspective, MIT Press."},{"key":"ref_2","unstructured":"Raginsky, M., Rakhlin, A., and Telgarsky, M. (2017, January 7\u201310). Non-convex learning via Stochastic Gradient Langevin Dynamics: A nonasymptotic analysis. Proceedings of the Conference on Learning Theory, Amsterdam, The Netherlands."},{"key":"ref_3","unstructured":"Welling, M., and Teh, Y.W. (July, January 28). Bayesian learning via stochastic gradient Langevin dynamics. Proceedings of the International Conference on Machine Learning, Washington, DC, USA."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3074","DOI":"10.3390\/e16063074","article-title":"Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions","volume":"16","author":"Livingstone","year":"2014","journal-title":"Entropy"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hartmann, C., Richter, L., Sch\u00fctte, C., and Zhang, W. (2017). Variational Characterization of Free Energy: Theory and Algorithms. Entropy, 19.","DOI":"10.3390\/e19110626"},{"key":"ref_6","unstructured":"Neal, R.M. (2004). Improving asymptotic variance of MCMC estimators: Non-reversible chains are better. arXiv."},{"key":"ref_7","unstructured":"Neklyudov, K., Welling, M., Egorov, E., and Vetrov, D. (2020, January 13\u201318). Involutive mcmc: A unifying framework. Proceedings of the International Conference on Machine Learning, Vienna, Austria."},{"key":"ref_8","unstructured":"Gao, X., Gurbuzbalaban, M., and Zhu, L. (2020, January 6\u201312). Breaking Reversibility Accelerates Langevin Dynamics for Non-Convex Optimization. Proceedings of the Advances in Neural Information Processing Systems, Online."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1982","DOI":"10.1214\/18-AOP1299","article-title":"Couplings and quantitative contraction rates for Langevin dynamics","volume":"47","author":"Eberle","year":"2019","journal-title":"Ann. Probab."},{"key":"ref_10","unstructured":"Gao, X., G\u00fcrb\u00fczbalaban, M., and Zhu, L. (2018). Global convergence of stochastic gradient Hamiltonian Monte Carlo for non-convex stochastic optimization: Non-asymptotic performance bounds and momentum-based acceleration. arXiv."},{"key":"ref_11","unstructured":"Cheng, X., Chatterji, N.S., Abbasi-Yadkori, Y., Bartlett, P.L., and Jordan, M.I. (2018). Sharp convergence rates for Langevin dynamics in the nonconvex setting. arXiv."},{"key":"ref_12","unstructured":"Chen, T., Fox, E., and Guestrin, C. (2014, January 21\u201326). Stochastic gradient hamiltonian monte carlo. Proceedings of the International conference on machine learning, Beijing, China."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1214\/aoap\/1177005371","article-title":"Accelerating gaussian diffusions","volume":"3","author":"Hwang","year":"1993","journal-title":"Ann. Appl. Probab."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1433","DOI":"10.1214\/105051605000000025","article-title":"Accelerating diffusions","volume":"15","author":"Hwang","year":"2005","journal-title":"Ann. Appl. Probab."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3522","DOI":"10.1016\/j.spa.2015.03.006","article-title":"Variance reduction for diffusions","volume":"125","author":"Hwang","year":"2015","journal-title":"Stoch. Process. Their Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1007\/s10955-016-1491-2","article-title":"Variance Reduction Using Nonreversible Langevin Samplers","volume":"163","author":"Duncan","year":"2016","journal-title":"J. Stat. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1098","DOI":"10.1007\/s10955-017-1906-8","article-title":"Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions","volume":"169","author":"Duncan","year":"2017","journal-title":"J. Stat. Phys."},{"key":"ref_18","unstructured":"Futami, F., Sato, I., and Sugiyama, M. (2020, January 13\u201318). Accelerating the diffusion-based ensemble sampling by non-reversible dynamics. Proceedings of the International Conference on Machine Learning, Vienna, Austria."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Bakry, D., Gentil, I., and Ledoux, M. (2013). Analysis and Geometry of Markov Diffusion Operators, Springer Science & Business Media.","DOI":"10.1007\/978-3-319-00227-9"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1051","DOI":"10.1051\/m2an\/2017044","article-title":"Spectral methods for Langevin dynamics and associated error estimates","volume":"52","author":"Roussel","year":"2018","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1809","DOI":"10.1214\/14-AOP908","article-title":"Poincar\u00e9 and logarithmic Sobolev inequalities by decomposition of the energy landscape","volume":"42","author":"Menz","year":"2014","journal-title":"Ann. Probab."},{"key":"ref_22","unstructured":"Liu, Q., Lee, J., and Jordan, M. (2016, January 24\u201326). A kernelized Stein discrepancy for goodness-of-fit tests. Proceedings of the International Conference on Machine Learning, New York, NY, USA."},{"key":"ref_23","unstructured":"Vempala, S., and Wibisono, A. (2019, January 8\u201314). Rapid convergence of the unadjusted langevin algorithm: Isoperimetry suffices. Proceedings of the Advances in Neural Information Processing Systems, Vancouver, BC, Canada."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s10955-013-0769-x","article-title":"Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion","volume":"152","author":"Nier","year":"2013","journal-title":"J. Stat. Phys."},{"key":"ref_25","unstructured":"Tripuraneni, N., Rowland, M., Ghahramani, Z., and Turner, R. (2017, January 6\u201311). Magnetic Hamiltonian Monte Carlo. Proceedings of the International Conference on Machine Learning, Sydney, Australia."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1137\/18M119896X","article-title":"Constructing sampling schemes via coupling: Markov semigroups and optimal transport","volume":"7","author":"Nusken","year":"2019","journal-title":"SIAM\/ASA J. Uncertain. Quantif."},{"key":"ref_27","unstructured":"Liu, Q., and Wang, D. (2016, January 5\u201310). Stein variational gradient descent: A general purpose bayesian inference algorithm. Proceedings of the Advances In Neural Information Processing Systems, Barcelona, Spain."},{"key":"ref_28","unstructured":"Zhang, J., Zhang, R., and Chen, C. (2018). Stochastic particle-optimization sampling and the non-asymptotic convergence theory. arXiv."},{"key":"ref_29","unstructured":"Wang, Y., and Li, W. (2020). Information Newton\u2019s flow: Second-order optimization method in probability space. arXiv."},{"key":"ref_30","unstructured":"Wibisono, A. (2018, January 6\u20139). Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem. Proceedings of the Conference On Learning Theory, Stockholm, Sweden."},{"key":"ref_31","first-page":"723","article-title":"A kernel two-sample test","volume":"13","author":"Gretton","year":"2012","journal-title":"J. Mach. Learn. Res."},{"key":"ref_32","unstructured":"Ding, N., Fang, Y., Babbush, R., Chen, C., Skeel, R.D., and Neven, H. (2014, January 8\u201311). Bayesian sampling using stochastic gradient thermostats. Proceedings of the Advances in neural information processing systems, Montreal, QC, Canada."},{"key":"ref_33","unstructured":"Patterson, S., and Teh, Y.W. (2013, January 5\u20138). Stochastic gradient Riemannian Langevin dynamics on the probability simplex. Proceedings of the Advances in Neural Information Processing Systems, Lake Tahoe, NV, USA."},{"key":"ref_34","unstructured":"Dua, D., and Graff, C. (2021, July 21). UCI Machine Learning Repository. Available online: http:\/\/archive.ics.uci.edu\/ml."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Villani, C. (2003). Optimal transportation, dissipative PDE\u2019s and functional inequalities. Optimal Transportation and Applications, Springer.","DOI":"10.1007\/978-3-540-44857-0_3"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1214\/ECP.v13-1352","article-title":"A simple proof of the Poincar\u00e9 inequality for a large class of probability measures including the log-concave case","volume":"13","author":"Bakry","year":"2008","journal-title":"Electron. Commun. Probab"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Nelson, E. (1967). Dynamical Theories of Brownian Motion, Princeton University Press.","DOI":"10.1515\/9780691219615"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Pavliotis, G.A. (2014). Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations, Springer.","DOI":"10.1007\/978-1-4939-1323-7"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1325","DOI":"10.1090\/S0002-9947-09-04939-3","article-title":"The behavior of the spectral gap under growing drift","volume":"362","author":"Franke","year":"2010","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1002\/cpa.21723","article-title":"Metastability of Nonreversible Random Walks in a Potential Field and the Eyring-Kramers Transition Rate Formula","volume":"71","author":"Landim","year":"2018","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1007\/s00205-018-1291-8","article-title":"Dirichlet\u2019s and Thomson\u2019s principles for non-selfadjoint elliptic operators with application to non-reversible metastable diffusion processes","volume":"231","author":"Landim","year":"2019","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Golub, G.H., and Van Loan, C.F. (2012). Matrix Computations, JHU Press.","DOI":"10.56021\/9781421407944"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1214\/aos\/1176342472","article-title":"Distinctness of the Eigenvalues of a Quadratic form in a Multivariate Sample","volume":"1","author":"Okamoto","year":"1973","journal-title":"Ann. Statist."},{"key":"ref_44","unstructured":"Petersen, K.B., and Pedersen, M.S. (2012). The Matrix Cookbook, Technical University of Denmark. Available online: http:\/\/www2.compute.dtu.dk\/pubdb\/pubs\/3274-full.html."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"3797","DOI":"10.1109\/TIT.2014.2320500","article-title":"R\u00e9nyi divergence and Kullback-Leibler divergence","volume":"60","author":"Harremos","year":"2014","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_46","unstructured":"Chewi, S., Le Gouic, T., Lu, C., Maunu, T., Rigollet, P., and Stromme, A. (2020, January 6\u201312). Exponential ergodicity of mirror-Langevin diffusions. Proceedings of the Advances in Neural Information Processing Systems, Online."},{"key":"ref_47","first-page":"331","article-title":"Weighted Csisz\u00e1r-Kullback-Pinsker inequalities and applications to transportation inequalities","volume":"Volume 14","author":"Bolley","year":"2005","journal-title":"Annales de la Facult\u00e9 des Sciences de Toulouse: Math\u00e9matiques"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1002\/cpa.3160360204","article-title":"Asymptotic evaluation of certain Markov process expectations for large time. IV","volume":"36","author":"Donsker","year":"1983","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1090\/conm\/353\/06431","article-title":"Logarithmic Sobolev inequalities and spectral gaps","volume":"353","author":"Carlen","year":"2004","journal-title":"Contemp. Math."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/993\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:37:57Z","timestamp":1760164677000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/993"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,30]]},"references-count":49,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["e23080993"],"URL":"https:\/\/doi.org\/10.3390\/e23080993","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,7,30]]}}}