On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry
This work addresses optimization challenges in machine learning and related fields that rely on SPD matrices, offering incremental improvements in algorithm robustness and convergence.
The paper tackled the problem of Riemannian optimization over symmetric positive definite matrices by comparing the Bures-Wasserstein geometry to the Affine-Invariant geometry, finding that the BW metric is more robust for ill-conditioned matrices and improves convergence rates due to its non-negative curvature.
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.