Nonpositive curvature, the variance functional, and the Wasserstein barycenter
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- by Young-Heon Kim and Brendan Pass PDF
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Abstract:
We show that a Riemannian manifold $M$ has nonpositive sectional curvature and is simply connected if and only if the variance functional on the space $P(M)$ of probability measures over $M$ is displacement convex. We then establish convexity over Wasserstein barycenters of the variance, and derive an inequality between the variance of the Wasserstein and linear barycenters of a probability measure on $P(M)$. These results are applied to invariant measures under isometry group actions, implying that the variance of the Wasserstein projection to the set of invariant measures is less than that of the $L^2$ projection to the same set.References
- Martial Agueh and Guillaume Carlier, Barycenters in the Wasserstein space, SIAM J. Math. Anal. 43 (2011), no. 2, 904–924. MR 2801182, DOI 10.1137/100805741
- Werner Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar, vol. 25, Birkhäuser Verlag, Basel, 1995. With an appendix by Misha Brin. MR 1377265, DOI 10.1007/978-3-0348-9240-7
- Jérôme Bertrand and Benoît Kloeckner, A geometric study of Wasserstein spaces: Hadamard spaces, J. Topol. Anal. 4 (2012), no. 4, 515–542. MR 3021775, DOI 10.1142/S1793525312500227
- Jérôme Bertrand and Benoît R. Kloeckner, A geometric study of Wasserstein spaces: an addendum on the boundary, Geometric science of information, Lecture Notes in Comput. Sci., vol. 8085, Springer, Heidelberg, 2013, pp. 405–412. MR 3126070, DOI 10.1007/978-3-642-40020-9_{4}4
- Jérôme Bertrand and Benoît R. Kloeckner, A geometric study of Wasserstein spaces: isometric rigidity in negative curvature, Int. Math. Res. Not. IMRN 5 (2016), 1368–1386. MR 3509929, DOI 10.1093/imrn/rnv177
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
- Dario Cordero-Erausquin, Robert J. McCann, and Michael Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math. 146 (2001), no. 2, 219–257. MR 1865396, DOI 10.1007/s002220100160
- Alfred Galichon and Nassif Ghoussoub, Variational representations for $N$-cyclically monotone vector fields, Pacific J. Math. 269 (2014), no. 2, 323–340. MR 3238477, DOI 10.2140/pjm.2014.269.323
- Nassif Ghoussoub and Bernard Maurey, Remarks on multi-marginal symmetric Monge-Kantorovich problems, Discrete Contin. Dyn. Syst. 34 (2014), no. 4, 1465–1480. MR 3121628, DOI 10.3934/dcds.2014.34.1465
- Nassif Ghoussoub and Abbas Moameni, Symmetric Monge-Kantorovich problems and polar decompositions of vector fields, Geom. Funct. Anal. 24 (2014), no. 4, 1129–1166. MR 3248482, DOI 10.1007/s00039-014-0287-2
- Nicola Gigli, On the inverse implication of Brenier-McCann theorems and the structure of $(\scr P_2(M),W_2)$, Methods Appl. Anal. 18 (2011), no. 2, 127–158. MR 2847481, DOI 10.4310/MAA.2011.v18.n2.a1
- Young-Heon Kim and Brendan Pass, Multi-marginal optimal transport on Riemannian manifolds, Amer. J. Math. 137 (2015), no. 4, 1045–1060. MR 3372314, DOI 10.1353/ajm.2015.0024
- Young-Heon Kim and Brendan Pass, Wasserstein barycenters over Riemannian manifolds, Adv. Math. 307 (2017), 640–683. MR 3590527, DOI 10.1016/j.aim.2016.11.026
- Thibaut Le Gouic and Jean-Michel Loubes, Existence and consistency of Wasserstein barycenters, Probab. Theory Related Fields 168 (2017), no. 3-4, 901–917. MR 3663634, DOI 10.1007/s00440-016-0727-z
- John Lott and Cédric Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), no. 3, 903–991. MR 2480619, DOI 10.4007/annals.2009.169.903
- Robert J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1997), no. 1, 153–179. MR 1451422, DOI 10.1006/aima.1997.1634
- Abbas Moameni, Invariance properties of the Monge-Kantorovich mass transport problem, Discrete Contin. Dyn. Syst. 36 (2016), no. 5, 2653–2671. MR 3485412, DOI 10.3934/dcds.2016.36.2653
- F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000), no. 2, 361–400. MR 1760620, DOI 10.1006/jfan.1999.3557
- Brendan Pass, Optimal transportation with infinitely many marginals, J. Funct. Anal. 264 (2013), no. 4, 947–963. MR 3004954, DOI 10.1016/j.jfa.2012.12.002
- Karl-Theodor Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), no. 1, 65–131. MR 2237206, DOI 10.1007/s11511-006-0002-8
- Karl-Theodor Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), no. 1, 133–177. MR 2237207, DOI 10.1007/s11511-006-0003-7
- Karl-Theodor Sturm, Probability measures on metric spaces of nonpositive curvature, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 357–390. MR 2039961, DOI 10.1090/conm/338/06080
- Cédric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. MR 1964483, DOI 10.1090/gsm/058
- Cédric Villani, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454, DOI 10.1007/978-3-540-71050-9
- Max-K. von Renesse and Karl-Theodor Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math. 58 (2005), no. 7, 923–940. MR 2142879, DOI 10.1002/cpa.20060
- D. A. Zaev, On the Monge-Kantorovich problem with additional linear constraints, Mat. Zametki 98 (2015), no. 5, 664–683 (Russian, with Russian summary); English transl., Math. Notes 98 (2015), no. 5-6, 725–741. MR 3438523, DOI 10.4213/mzm10896
Additional Information
- Young-Heon Kim
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2 Canada
- MR Author ID: 615856
- ORCID: 0000-0001-6920-603X
- Email: yhkim@math.ubc.ca
- Brendan Pass
- Affiliation: Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, Alberta, T6G 2G1 Canada
- MR Author ID: 963854
- Email: pass@ualberta.ca.
- Received by editor(s): April 2, 2019
- Received by editor(s) in revised form: August 24, 2019
- Published electronically: January 13, 2020
- Additional Notes: The first author was supported in part by Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grants 371642-09 and 2014-0544, as well as Alfred P. Sloan research fellowship 2012-2016.
The second author is pleased to acknowledge the support of a University of Alberta start-up grant and National Sciences and Engineering Research Council of Canada Discovery Grant numbers 412779-2012 and 04658-2018. - Communicated by: Guofang Wei
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 1745-1756
- MSC (2010): Primary 53C21; Secondary 49Q99
- DOI: https://doi.org/10.1090/proc/14840
- MathSciNet review: 4069211