Abstract
We investigate contraction of the Wasserstein distances on \(\mathbb {R}^{d}\) under Gaussian smoothing. It is well known that the heat semigroup is exponentially contractive with respect to the Wasserstein distances on manifolds of positive curvature; however, on flat Euclidean space—where the heat semigroup corresponds to smoothing the measures by Gaussian convolution—the situation is more subtle. We prove precise asymptotics for the 2-Wasserstein distance under the action of the Euclidean heat semigroup, and show that, in contrast to the positively curved case, the contraction rate is always polynomial, with exponent depending on the moment sequences of the measures. We establish similar results for the p-Wasserstein distances for p≠ 2 as well as the χ2 divergence, relative entropy, and total variation distance. Together, these results establish the central role of moment matching arguments in the analysis of measures smoothed by Gaussian convolution.
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References
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2005). ISBN 978-3-7643-2428-5; 3-7643-2428-7
Ambrosio, L., Stra, F., Trevisan, D.: A PDE approach to a 2-dimensional matching problem. Probab. Theory Relat. Fields 173(1-2), 433–477 (2019). ISSN 0178-8051. https://doi.org/10.1007/s00440-018-0837-x
Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators, volume 348 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Cham (2014). ISBN 978-3-319-00226-2; 978-3-319-00227-9. https://doi.org/10.1007/978-3-319-00227-9
Bandeira, A.S., Niles-Weed, J., Rigollet, P.: Optimal rates of estimation for multi-reference alignment. Math. Stat. Learn. To appear (2020)
Bolley, F., Gentil, I., Guillin, A.: Dimensional contraction via Markov transportation distance. J. London Math. Soc. Second Series 90(1), 309–332 (2014). ISSN 0024-6107. https://doi.org/10.1112/jlms/jdu027
Bolley, F., Gentil, I., Guillin, A.: Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities. Ann. Probab. 46(1), 261–301 (2018). ISSN 0091-1798. https://doi.org/10.1214/17-AOP1184
Brasco, L.: A survey on dynamical transport distances. J. Math. Sci. 181(6), 755–781 (2012)
Caracciolo, S., Lucibello, C., Parisi, G., Sicuro, G.: Scaling hypothesis for the euclidean bipartite matching problem. Phys. Rev. E 90, 012118 (2014). https://doi.org/10.1103/PhysRevE.90.012118
Chewi, S., Maunu, T., Rigollet, P., Stromme, A.J.: Gradient descent algorithms for Bures-W,asserstein barycenters. arXiv:2001.01700 (2020)
Csiszár, I.: Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten. Magyar Tud. Akad. Mat. Kutató Int. Kö,zl. 8, 85–108 (1963)
Eberle, A.: Reflection coupling and Wasserstein contractivity without convexity. Comptes Rendus Mathématique. Académie des Sciences. Paris 349 (19–20), 1101–1104 (2011). ISSN 1631-073X. https://doi.org/10.1016/j.crma.2011.09.003
Eberle, A.: Reflection couplings and contraction rates for diffusions. Probab. Theory Relat. Field 166(3-4), 851–886 (2016). ISSN 0178-8051. https://doi.org/10.1007/s00440-015-0673-1
Giaquinta, M., Martinazzi, L.: An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs. Springer Science & Business Media, New York (2013)
Givens, C.R., Shortt, R.M., et al.: A class of Wasserstein metrics for probability distributions. Michigan Math. J. 31(2), 231–240 (1984)
Goldfeld, Z., Greenewald, K.: Gaussian-smooth optimal transport: Metric structure and statistical efficiency. arXiv:2001.09206 (2020)
Goldfeld, Z., Niles-Weed, J., Polyanskiy, Y.: Convergence of smoothed empirical measures with applications to entropy estimation. IEEE Trans. Inform. Theory, Greenewald, K. To appear (2020)
Ledoux, M.: The Concentration of Measure Phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2001). ISBN 0-8218-2864-9
Liese, F., Vajda, I.: On divergences and informations in statistics and information theory. IEEE Trans. Inform. Theory 52(10), 4394–4412 (2006). ISSN 0018-9448. https://doi.org/10.1109/TIT.2006.881731
Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. Second Series 169(3), 903–991 (2009). ISSN 0003-486X. https://doi.org/10.4007/annals.2009.169.903
Luo, D., Wang, J.: Exponential convergence in Lp-wasserstein distance for diffusion processes without uniformly dissipative drift. Mathematische Nachrichten 289(14-15), 1909–1926 (2016)
Marton, K.: Bounding \(\overline d\)-distance by informational divergence: a method to prove measure concentration. Ann. Probab. 24 (2), 857–866 (1996a). ISSN 0091-1798. https://doi.org/10.1214/aop/1039639365
Marton, K.: A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6(3), 556–571 (1996b). ISSN 1016-443X. https://doi.org/10.1007/BF02249263
McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–179 (1997). ISSN 0001-8708. https://doi.org/10.1006/aima.1997.1634
Moser, J.: On the volume elements on a manifold. Trans. Amer. Math. Soc. 120, 286–294 (1965). ISSN 0002-9947. https://doi.org/10.2307/1994022
Nualart, D.: The Malliavin Calculus and Related Topics. Springer, Berlin (2006)
Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Part. Differ. Eq. 26(1-2), 101–174 (2001). ISSN 0360-5302. https://doi.org/10.1081/PDE-100002243
Otto, F., Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173(2), 361–400 (2000). ISSN 0022-1236. https://doi.org/10.1006/jfan.1999.3557
Peyre, R.: Comparison between W2 distance and \(\dot ~\mathrm {H}^{-1}\) norm, and localization of Wasserstein distance. ESAIM Control Optim. Calc. Var. 24(4), 1489–1501 (2018). https://doi.org/10.1051/cocv/2017050. ISSN 1292-8119
Pisier, G.: Probabilistic methods in the geometry of banach spaces. In: Probability and Analysis, pp 167–241. Springer (1986)
Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Mathematica 196(1), 133–177 (2006a). https://doi.org/10.1007/s11511-006-0003-7. ISSN 0001-5962
Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Mathematica 196(1), 65–131 (2006b). https://doi.org/10.1007/s11511-006-0002-8. ISSN 0001-5962
Talagrand, M.: Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6(3), 587–600 (1996). https://doi.org/10.1007/BF02249265. ISSN 1016-443X
Villani, C.: Optimal Transport: Old and New, vol. 338. Springer Science & Business Media, New York (2009)
Villani, C.: Synthetic theory of Ricci curvature bounds. Jpn. J. Math. 11(2), 219–263 (2016). https://doi.org/10.1007/s11537-016-1531-3. ISSN 0289-2316
von Renesse, M.-K., Sturm, K.-T.: Transport inequalities, gradient estimates, entropy and Ricci curvature. Commun. Pure Appl. Math. 58(7), 923–940 (2005). https://doi.org/10.1002/cpa.20060. https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.20060
Wang, F.-Y.: Analysis for Diffusion Processes on Riemannian Manifolds, volume 18 of Advanced Series on Statistical Science & Applied Probability,. World Scientific Publishing Co. Pte. Ltd., Hackensack (2014). ISBN 978-981-4452-64-9
Wang, F. -Y.: Exponential contraction in wasserstein distances for diffusion semigroups with negative curvature. arXiv:1603.05749 (2016)
Weed, J.: Sharper rates for estimating differential entropy under gaussian convolutions. Massachusetts Institute of Technology, MIT, Tech. Rep (2018)
Wu, Y., Yang, P.: Optimal estimation of gaussian mixtures via denoised method of moments. arXiv:1807.07237 (2018)
Zhang, S.-Q.: Exponential convergence in wasserstein distance for diffusion semigroups with irregular drifts. arXiv:1812.10190 (2018)
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JNW gratefully acknowledges the support of the Institute for Advanced Study, where a portion of this research was conducted.
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Chen, HB., Niles-Weed, J. Asymptotics of Smoothed Wasserstein Distances. Potential Anal 56, 571–595 (2022). https://doi.org/10.1007/s11118-020-09895-9
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DOI: https://doi.org/10.1007/s11118-020-09895-9