Riemannian Gradient Method for Wasserstein Barycenters under the Affine-Invariant Geometry

We study a Riemannian gradient method for the $L_2$-Wasserstein least squares problem of Gaussian measures under the affine-invariant geometry. The variable of $L_2$-Wasserstein least squares problem lies in the set of positive definite matrices, which, equipped with the affine-invariant metric, forms a Hadamard manifold. The same set with usual Euclidean metric is also a Hadamard manifold, with constant sectional curvature equal to 0. Hence, the gradient descent method proposed in [S. Kum, S. Yun, J. Korean Math Soc. $\mathbf{56}$ (2019)] can be considered as a Riemannian gradient method with respect to usual Euclidean inner product. This method is known to have a sublinear convergence rate and requires a singular value decomposition at each iteration, which is computationally expensive. In this paper, we adapt the Riemannian gradient method under the affine-invariant geometry for solving the $L_2$-Wasserstein least squares and prove its local linear convergence. This method does not require a singular value decomposition. We numerically show that the proposed method is more efficient than the gradient descent method mentioned.