# Optimization over covariance matrices with a parameterized metric

arXiv: https://arxiv.org/abs/2609.17089 · HTML: https://mlsanity.com/paper/2609.17089/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 26 references that are in the map.
- First posted on arXiv 15 Sep 2026.

## Builds on (most cited first, 6 of 6)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry](https://mlsanity.com/paper/2110.10464/) (2021)
- [Riemannian optimization with a preconditioning scheme on the generalized Stiefel manifold](https://mlsanity.com/paper/1902.01635/) (2019)
- [Optimization on product manifolds under a preconditioned metric](https://mlsanity.com/paper/2306.08873/) (2023)
- [Riemannian Optimization over Symmetric Positive Definite Matrices with the Alpha-Procrustes Geometry](https://mlsanity.com/paper/2605.00396/) (2026)

## Abstract (arXiv)

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.