Optimization over covariance matrices with a parameterized metric

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 XpLXq+XqLXp=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 conditioning of the Riemannian Hessian at the solution. We show that it obeys a lower bound that depends on (p,q) only through the exponent r=p+q. When the Euclidean Hessian is a pure power that mixes no eigendirections, the member p=q=r/2 attains that bound, and a closed-form criterion identifies the other members that do. We discuss ways to tune r for a given problem. Experiments on real covariance data confirm the predicted conditioning and the benefit of tuning r. A task covariance example shows a further gain from tuning the shape.