This work uncovers variational principles behind symmetrizing the Bregman divergences induced by generic mirror maps over the cone of positive definite matrices. We show that computing the canonical means for this symmetrization can be posed as minimizing the desired symmetrized divergences over a set of mean functionals defined axiomatically to satisfy certain properties. For the forward symmetrization, we prove that the arithmetic mean over the primal space is canonical for any mirror map over the positive definite cone. For the reverse symmetrization, we show that the canonical mean is the arithmetic mean over the dual space, pulled back to the primal space. Applying this result to three common mirror maps used in practice, we show that the canonical means for reverse symmetrization, in those cases, turn out to be the arithmetic, log-Euclidean and harmonic means. Our results improve understanding of existing symmetrization practices in the literature, and can be seen as a navigational chart to help decide which mean to use when.

翻译：本文揭示了在正定矩阵锥上，由通用镜像映射诱导的Bregman散度在对称化过程中的变分原理。我们证明，计算该对称化过程的典范均值可归结为：在一组依据公理定义并满足特定性质的均值泛函集上，最小化所需对称化散度。对于正向对称化，我们证明了在正定矩阵锥上，对于任意镜像映射，原始空间中的算术均值具有典范性。对于反向对称化，我们表明典范均值是原始空间中回溯的对偶空间算术均值。将此结果应用于实践中三种常见的镜像映射，我们证明在这些情形下，反向对称化对应的典范均值分别为算术均值、对数欧氏均值与调和均值。我们的研究深化了现有文献中对称化实践的理解，并可作为帮助决策在何时使用何种均值的导航图。