Wasserstein Riemannian Geometry of Positive-definite Matrices∗

The Wasserstein distance on multivariate non-degenerate Gaussian densities is a Riemannian distance. After reviewing the properties of the distance and the metric geodesic, we derive an explicit form of the Riemannian metrics on positive-definite matrices and compute its tensor form with respect to the trace scalar product. The tensor is a matrix, which is the solution of a Lyapunov equation. We compute explicit form for the Riemannian exponential, the normal coordinates charts, the Riemannian gradient, and discuss the gradient flow of the entropy. Finally, the Levi-Civita covariant derivative is computed in matrix form together with the differential equation for the parallel transport. While all computations are given in matrix form, notheless we discuss the use of a special moving frame. Applications are briefly discussed.