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Transportation onto log-Lipschitz perturbations (2305.03786v2)
Published 5 May 2023 in math.PR, math.AP, and math.FA
Abstract: We establish sufficient conditions for the existence of globally Lipschitz transport maps between probability measures and their log-Lipschitz perturbations, with dimension-free bounds. Our results include Gaussian measures on Euclidean spaces and uniform measures on spheres as source measures. More generally, we prove results for source measures on manifolds satisfying strong curvature assumptions. These seem to be the first examples of dimension-free Lipschitz transport maps in non-Euclidean settings, which are moreover sharp on the sphere. We also present some applications to functional inequalities, including a new dimension-free Gaussian isoperimetric inequality for log-Lipschitz perturbations of the standard Gaussian measure. Our proofs are based on the Langevin flow construction of transport maps of Kim and Milman.
- Logarithmic Sobolev inequalities and spectral gaps: perturbation theory. J. Funct. Anal., 126(2):448–475, 1994.
- Analysis and geometry of Markov diffusion operators, volume 348 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Cham, 2014.
- Lévy-Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math., 123(2):259–281, 1996.
- Functional inequalities for Gaussian convolutions of compactly supported measures: explicit bounds and dimension dependence. Bernoulli, 24(1):333–353, 2018.
- Mass transport and variants of the logarithmic Sobolev inequality. J. Geom. Anal., 18(4):921–979, 2008.
- The Friedland-Hayman inequality and Caffarelli’s contraction theorem. J. Math. Phys., 62(10):Paper No. 101504, 11, 2021.
- Jean-Michel Bismut. Martingales, the Malliavin calculus and hypoellipticity under general Hörmander’s conditions. Z. Wahrsch. Verw. Gebiete, 56(4):469–505, 1981.
- Sergey G. Bobkov. An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probab., 25(1):206–214, 1997.
- Diffusion Schrödinger bridge with applications to score-based generative modeling. In A. Beygelzimer, Y. Dauphin, P. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, 2021.
- Luis A. Caffarelli. Monotonicity properties of optimal transportation and the FKG and related inequalities. Comm. Math. Phys., 214(3):547–563, 2000.
- Some inequalities on Riemannian manifolds linking entropy, Fisher information, Stein discrepancy and Wasserstein distance. arXiv preprint arXiv:2108.12755, 2021.
- An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities. arXiv preprint arXiv:2203.04954, 2022.
- Moser inequalities in Gauss space. Math. Ann., 377(3-4):1265–1312, 2020.
- Bounds on optimal transport maps onto log-concave measures. J. Differential Equations, 271:1007–1022, 2021.
- Lipschitz changes of variables between perturbations of log-concave measures. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 17(4):1491–1519, 2017.
- Dario Cordero-Erausquin. Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal., 161(3):257–269, 2002.
- Formulae for the derivatives of heat semigroups. J. Funct. Anal., 125(1):252–286, 1994.
- Correction to: Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula. Probab. Theory Related Fields, 175(3-4):1177–1181, 2019.
- Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula. Probab. Theory Related Fields, 174(3-4):945–979, 2019.
- A proof of the Caffarelli contraction theorem via entropic regularization. Calc. Var. Partial Differential Equations, 59(3):Paper No. 96, 18, 2020.
- Alessio Figalli. Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper]. Number 332, pages Exp. No. 1009, ix, 341–368. 2010. Séminaire Bourbaki. Volume 2008/2009. Exposés 997–1011.
- Measuring sample quality with diffusions. Ann. Appl. Probab., 29(5):2884–2928, 2019.
- Gilles Hargé. A particular case of correlation inequality for the Gaussian measure. Ann. Probab., 27(4):1939–1951, 1999.
- Denoising diffusion probabilistic models. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 6840–6851. Curran Associates, Inc., 2020.
- Elton P. Hsu. Stochastic analysis on manifolds, volume 38 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.
- Exponential integrability for log-concave measures. arXiv preprint arXiv:2004.09704, 2020.
- Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991.
- A generalization of Caffarelli’s contraction theorem via (reverse) heat flow. Math. Ann., 354(3):827–862, 2012.
- Spectral monotonicity under gaussian convolution. Ann. Fac. Sci. Toulouse Math, To appear.
- Alexander V Kolesnikov. Mass transportation and contractions. arXiv preprint arXiv:1103.1479, 2011.
- Alexander V. Kolesnikov. On Sobolev regularity of mass transport and transportation inequalities. Theory Probab. Appl., 57(2):243–264, 2013.
- The flow map of the Fokker-Planck equation does not provide optimal transport. Appl. Math. Lett., 133:Paper No. 108225, 7, 2022.
- Dan Mikulincer. A CLT in Stein’s distance for generalized Wishart matrices and higher-order tensors. Int. Math. Res. Not. IMRN, (10):7839–7872, 2022.
- On the Lipschitz properties of transportation along heat flows. GAFA Seminar Notes, To appear.
- Emanuel Milman. Spectral estimates, contractions and hypercontractivity. J. Spectr. Theory, 8(2):669–714, 2018.
- Emanuel Milman. Reverse Hölder inequalities for log-Lipschitz functions. Pure Appl. Funct. Anal., 8(1):297–310, 2023.
- Joe Neeman. Lipschitz changes of variables via heat flow. arXiv preprint arXiv:2201.03403, 2022.
- F. Otto and C. Villani. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal., 173(2):361–400, 2000.
- Yair Shenfeld. Exact renormalization groups and transportation of measures. arXiv preprint arXiv:2205.01642, 2022.
- Anastasiya Tanana. Comparison of transport map generated by heat flow interpolation and the optimal transport Brenier map. Commun. Contemp. Math., 23(6):Paper No. 2050025, 7, 2021.
- James Thompson. Approximation of Riemannian measures by Stein’s method. arXiv preprint arXiv:2001.09910, 2020.
- Cédric Villani. Optimal transport: old and new, volume 338. Springer Science & Business Media, 2008.
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