Wasserstein Barycenters over Riemannian manifolds

[1]  R. Bishop,et al.  Geometry of Manifolds , 1964 .

[2]  Richard A. Vitale,et al.  The Brunn-Minkowski inequality for random sets , 1990 .

[3]  M. Émery,et al.  Sur le barycentre d'une probabilité dans une variété , 1991 .

[4]  Svetlozar T. Rachev,et al.  Maximum submatrix traces for positive definite matrices , 1993 .

[5]  M. Knott,et al.  On a generalization of cyclic monotonicity and distances among random vectors , 1994 .

[6]  R. McCann.  A convexity theory for interacting gases and equilibrium crystals , 1994 .

[7]  W. Gangbo,et al.  The geometry of optimal transportation , 1996 .

[8]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

[9]  L. Rüschendorf,et al.  On Optimal Multivariate Couplings , 1997 .

[10]  W. Gangbo,et al.  Optimal maps for the multidimensional Monge-Kantorovich problem , 1998 .

[11]  S. Graf,et al.  Foundations of Quantization for Probability Distributions , 2000 .

[12]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[13]  R. McCann,et al.  A Riemannian interpolation inequality à la Borell, Brascamp and Lieb , 2001 .

[14]  R. McCann.  Polar factorization of maps on Riemannian manifolds , 2001 .

[15]  Frank Morgan,et al.  Hexagonal Economic Regions Solve the Location Problem , 2002, Am. Math. Mon..

[16]  Karl-Theodor Sturm,et al.  Probability Measures on Metric Spaces of Nonpositive Curvature , 2003 .

[17]  C. Villani.  Topics in Optimal Transportation , 2003 .

[18]  C. Villani,et al.  Ricci curvature for metric-measure spaces via optimal transport , 2004, math/0412127.

[19]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[20]  N. Trudinger,et al.  Regularity of Potential Functions of the Optimal Transportation Problem , 2005 .

[21]  S. Bianchini.  On the Euler-Lagrange Equation for a Variational Problem , 2006 .

[22]  Karl-Theodor Sturm,et al.  On the geometry of metric measure spaces. II , 2006 .

[23]  Karl-Theodor Sturm,et al.  On the geometry of metric measure spaces , 2006 .

[24]  N. Trudinger,et al.  On the second boundary value problem for Monge-Ampère type equations and optimal transportation , 2006, math/0601086.

[25]  R. McCann,et al.  Continuity, curvature, and the general covariance of optimal transportation , 2007, 0712.3077.

[26]  Existence and uniqueness of optimal maps on Alexandrov spaces , 2007, 0705.0437.

[27]  Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds , 2007, 0709.1653.

[28]  Robert J. McCann,et al.  Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular) , 2008, 0806.0351.

[29]  A. Figalli,et al.  Absolute continuity of Wasserstein geodesics in the Heisenberg group , 2008 .

[30]  C. Villani.  Optimal Transport: Old and New , 2008 .

[31]  Filippo Santambrogio.  Absolute continuity and summability of transport densities: simpler proofs and new estimates , 2009 .

[32]  G. Loeper.  On the regularity of solutions of optimal transportation problems , 2009 .

[33]  A. Figalli,et al.  Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2 , 2009 .

[34]  Locally nearly spherical surfaces are almost-positively $c$-curved , 2010, 1009.3586.

[35]  G. Carlier,et al.  Matching for teams , 2010 .

[36]  S. Bianchini,et al.  On the Euler–Lagrange equation for a variational problem: the general case II , 2010 .

[37]  Guillaume Carlier,et al.  Barycenters in the Wasserstein Space , 2011, SIAM J. Math. Anal..

[38]  Alessio Figalli,et al.  When is multidimensional screening a convex program? , 2009, J. Econ. Theory.

[39]  Julien Rabin,et al.  Wasserstein Barycenter and Its Application to Texture Mixing , 2011, SSVM.

[40]  G. Loeper.  Regularity of Optimal Maps on the Sphere: the Quadratic Cost and the Reflector Antenna , 2013, 1301.6229.

[41]  Brendan Pass.  Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions , 2012, 1210.7372.

[42]  Brendan Pass.  Optimal transportation with infinitely many marginals , 2012, 1206.5515.

[43]  Jérémie Bigot,et al.  Consistent estimation of a population barycenter in the Wasserstein space , 2013 .

[44]  Brendan Pass,et al.  Multi-marginal optimal transport on Riemannian manifolds , 2013, 1303.6251.

[45]  Philippe Delanoë,et al.  Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved , 2013, Canadian Journal of Mathematics.

[46]  K. Kuwae.  Jensen’s inequality on convex spaces , 2014 .

[47]  Brendan Pass.  Multi-marginal optimal transport: theory and applications , 2014, 1406.0026.

[48]  F. Santambrogio.  Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling , 2015 .