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Optimization over covariance matrices with a parameterized metric

arXiv · AI, language, vision and robotics · article · Sep 15, 2026 · UTC

The choice of Riemannian metric can strongly influence the convergence of gradient-based optimization over covariance matrices. Euclidean, Bures-Wasserstein and affine-invariant metrics are common choices, but their relative effectiveness depends on the objective. We introduce a two-parameter family defined by $X^{p}LX^{q}+X^{q}LX^{p}=U$, solved for $L$ at each tangent vector $U$, that contains all three as exact members, at $(0,0)$, $(1,0)$ and $(1,1)$, and extends past them. We treat the choice of member as a particular way of preconditioning for a given problem. To this end, we analyze the

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