  In this paper, we comparatively analyze the Bures-Wasserstein (BW) geometry
with the popular Affine-Invariant (AI) geometry for Riemannian optimization on
the symmetric positive definite (SPD) matrix manifold. Our study begins with an
observation that the BW metric has a linear dependence on SPD matrices in
contrast to the quadratic dependence of the AI metric. We build on this to show
that the BW metric is a more suitable and robust choice for several Riemannian
optimization problems over ill-conditioned SPD matrices. We show that the BW
geometry has a non-negative curvature, which further improves convergence rates
of algorithms over the non-positively curved AI geometry. Finally, we verify
that several popular cost functions, which are known to be geodesic convex
under the AI geometry, are also geodesic convex under the BW geometry.
Extensive experiments on various applications support our findings.
