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 $X^{p}LX^{q}+X^{q}LX^{p}=U$, 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 conditioning of the Riemannian Hessian at the solution. We show that it obeys a lower bound that ...
| Authors | Yibang Li, Bamdev Mishra, Pratik Jawanpuria et al. |
|---|---|
| Primary category | math.OC |
| Published | 15 September 2026 |
| Last updated at source | 15 September 2026 |
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