On Wasserstein geometry of the space of Gaussian measures

The space of Gaussian measures on a Euclidean space is geodesically convex in the $L^2$-Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By restricting to the space of Gaussian measures inside the $L^2$-Wasserstein space, we manage to provide detailed descriptions of the $L^2$-Wasserstein geometry from a Riemannian geometric viewpoint. We first construct a Riemannian metric which induces the $L^2$-Wasserstein distance. Then we obtain a formula for the sectional curvatures of the space of Gaussian measures, which is written out in terms of the eigenvalues of the covariance matrix.

[1]  Y. Brenier.  Polar Factorization and Monotone Rearrangement of Vector-Valued Functions , 1991 .

[2]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[3]  Shin-ichi Ohta,et al.  Gradient flows on Wasserstein spaces over compact Alexandrov spaces , 2009 .

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

[5]  I. Olkin,et al.  The distance between two random vectors with given dispersion matrices , 1982 .

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

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

[8]  F. Götze.  Differential-geometrical methods in statistics. Lecture notes in statistics - A. Shun-ichi. , 1987 .

[9]  Shun-ichi Amari,et al.  Differential-geometrical methods in statistics , 1985 .

[10]  D. Dowson,et al.  The Fréchet distance between multivariate normal distributions , 1982 .

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

[12]  D. Burago,et al.  A Course in Metric Geometry , 2001 .

[13]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[14]  B. O'neill,et al.  The fundamental equations of a submersion. , 1966 .

[15]  C. Givens,et al.  A class of Wasserstein metrics for probability distributions. , 1984 .

[16]  M. Knott,et al.  On the optimal mapping of distributions , 1984 .

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

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

[19]  C. Villani,et al.  Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media , 2006 .