×

Wasserstein geometry of Gaussian measures. (English) Zbl 1245.60013

In this paper the geometric structure of Gaussian measures is considered. The space \(\mathcal{N}^d\) of Gaussian measures on \(\mathbb{R}^d\) is a space of finite dimension, and this allows to write down the explicit Riemannian metric, which, in turn, induces the \(L^2\)-Wasserstein distance function.
After introducing the \(L^2\)-Wasserstein geometry, results concerning the \(L^2\)-Wasserstein geometry on \(\mathcal{N}^d\) are determined, and the \(L^2\)-Wasserstein metric is analyzed. Then the completion \(\overline{\mathcal{N}_0^d}\) of the space \(\mathcal{N}_0^d\) (consisting of Gaussian measures with mean \(0\)) is studied as an Alexandrov space. Stratification and tangent cones are investigated. In particular, it turns out that the singular set is stratified according to the dimension of the support of the Gaussian measures, providing an explicit nontrivial example of Alexandrov space with extremal sets.

MSC:

60D05 Geometric probability and stochastic geometry
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
PDF BibTeX XML Cite
Full Text: Euclid

References:

[1] S. Amari: Differential-Geometrical Methods in Statistics, Lecture Notes in Statistics 28 , Springer, New York, 1985. · Zbl 0559.62001
[2] L. Ambrosio, N. Gigli and G. Savaré: Gradient Flows in Metric Spaces and in the Space of Probability Measures, second edition, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 2008. · Zbl 1145.35001
[3] P. Billingsley: Probability and Measure, third edition, Wiley Series in Probability and Mathematical Statistics, Wiley, New York, 1995. · Zbl 0822.60002
[4] Y. Brenier: Polar factorization and monotone rearrangement of vector-valued functions , Comm. Pure Appl. Math. 44 (1991), 375-417. · Zbl 0738.46011
[5] D. Burago, Y. Burago and S. Ivanov: A Course in Metric Geometry, Graduate Studies in Mathematics 33 , Amer. Math. Soc., Providence, RI, 2001. · Zbl 0981.51016
[6] Y. Burago, M. Gromov and G. Perelman: A.D. Aleksandrov spaces with curvatures bounded below , Uspekhi Mat. Nauk 47 (1992), 3-51, 222.
[7] D.C. Dowson and B.V. Landau: The Fréchet distance between multivariate normal distributions , J. Multivariate Anal. 12 (1982), 450-455. · Zbl 0501.62038
[8] S. Gallot, D. Hulin and J. Lafontaine: Riemannian Geometry, third edition, Universitext, Springer, Berlin, 2004.
[9] W. Gangbo and R.J. McCann: The geometry of optimal transportation , Acta Math. 177 (1996), 113-161. · Zbl 0887.49017
[10] \begingroup C.R. Givens and R.M. Shortt: A class of Wasserstein metrics for probability distributions , Michigan Math. J. 31 (1984), 231-240. \endgroup · Zbl 0582.60002
[11] M. Knott and C.S. Smith: On the optimal mapping of distributions , J. Optim. Theory Appl. 43 (1984), 39-49. · Zbl 0519.60010
[12] J. Lott: Some geometric calculations on Wasserstein space , Comm. Math. Phys. 277 (2008), 423-437. · Zbl 1144.58007
[13] J. Lott: Optimal transport and Perelman’s reduced volume , Calc. Var. Partial Differential Equations 36 (2009), 49-84. · Zbl 1171.53318
[14] J. Lott and C. Villani: Weak curvature conditions and functional inequalities , J. Funct. Anal. 245 (2007), 311-333. · Zbl 1119.53028
[15] J. Lott and C. Villani: Ricci curvature for metric-measure spaces via optimal transport , Ann. of Math. (2) 169 (2009), 903-991. · Zbl 1178.53038
[16] R.J. McCann: A convexity principle for interacting gases , Adv. Math. 128 (1997), 153-179. · Zbl 0901.49012
[17] R.J. McCann: Polar factorization of maps on Riemannian manifolds , Geom. Funct. Anal. 11 (2001), 589-608. · Zbl 1011.58009
[18] R.J. McCann and P.M. Topping: Ricci flow, entropy and optimal transportation , Amer. J. Math. 132 (2010), 711-730. · Zbl 1203.53065
[19] I. Olkin and F. Pukelsheim: The distance between two random vectors with given dispersion matrices , Linear Algebra Appl. 48 (1982), 257-263. · Zbl 0527.60015
[20] B. O’Neill: The fundamental equations of a submersion , Michigan Math. J. 13 (1966), 459-469. · Zbl 0145.18602
[21] F. Otto: The geometry of dissipative evolution equations: the porous medium equation , Comm. Partial Differential Equations 26 (2001), 101-174. · Zbl 0984.35089
[22] G. Perelman: Elements of Morse theory on Aleksandrov spaces , Algebra i Analiz 5 (1993), 232-241. · Zbl 0815.53072
[23] G. Perelman: The entropy formula for the ricci flow and its geometric applications , · Zbl 1130.53001
[24] G. Perelman and A.M. Petrunin: Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem , Algebra i Analiz 5 (1993), 242-256. · Zbl 0802.53019
[25] A. Petrunin: Semiconcave functions in Alexandrov’s geometry ; in Surveys in Differential Geometry 11, Surv. Differ. Geom. 11 , Int. Press, Somerville, MA, 137-201, 2007. · Zbl 1166.53001
[26] K.-T. Sturm: On the geometry of metric measure spaces , I, Acta Math. 196 (2006), 65-131. · Zbl 1105.53035
[27] K.-T. Sturm: On the geometry of metric measure spaces , II, Acta Math. 196 (2006), 133-177. · Zbl 1106.53032
[28] A. Takatsu and T. Yokota, Cone structure of \(\mathit{L}^{2}\)-Wasserstein spaces , · Zbl 1253.28001
[29] P. Topping: \(\mathrsfs{L}\)-optimal transportation for Ricci flow , J. Reine Angew. Math. 636 (2009), 93-122. · Zbl 1187.53072
[30] C. Villani: Topics in Optimal Transportation, Graduate Studies in Mathematics 58 , Amer. Math. Soc., Providence, RI, 2003. · Zbl 1106.90001
[31] C. Villani: Optimal Transport, Grundlehren der Mathematischen Wissenschaften 338 , Springer, Berlin, 2009.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.