Optimization over covariance matrices with a parameterized metric
Abstract
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.
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BibTeX
@misc{li2026optimization,
title = {{Optimization over covariance matrices with a parameterized metric}},
author = {Yibang Li and Bamdev Mishra and Pratik Jawanpuria and Cyrus Mostajeran},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17089},
eprint = {2609.17089},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17089},
}
RIS
TY - UNPB
TI - Optimization over covariance matrices with a parameterized metric
AU - Yibang Li
AU - Bamdev Mishra
AU - Pratik Jawanpuria
AU - Cyrus Mostajeran
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17089
UR - https://arxiv.org/abs/2609.17089
AB - 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^pLX^q+X^qLX^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 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.
ER -