Optimization over covariance matrices with a parameterized metric

2026-09-15

AI Summary

研究通过引入一个两参数家族来优化协方差矩阵，该方法能包含并扩展常见的度量方式，通过调整参数以改善特定问题的优化效果。

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 $X^{p}LX^{q}+X^{q}LX^{p}=U$, 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 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.

Problem

• Riemannian metric
• covariance matrices
• gradient-based optimization
• convergence
• conditioning

Innovation

• two-parameter family
• Riemannian metric
• covariance matrices optimization
• conditioning of Riemannian Hessian
• parameter tuning