揭示正定矩阵上Bregman散度对称化时该用哪种均值及其原理。
Symmetrizing Bregman Divergence on the Cone of Positive Definite Matrices: Which Mean to Use and Why
- 通过变分原理确定对称化所需的最优均值形式。
- 前向对称化用原空间算术平均，反向对称化用对偶空间算术平均。
- 为实际应用中选择均值提供理论依据，适合矩阵分析研究者。
本文揭示了在正定矩阵锥上，由通用镜映射诱导的Bregman散度对称化的变分原理。我们证明，计算此类对称化对应的规范均值，可转化为在满足特定公理性质的均值函数族上最小化目标对称散度的问题。对于前向对称化，我们证明任意镜映射下，原空间的算术平均是规范均值；对于反向对称化，规范均值为对偶空间的算术平均，经拉回至原空间所得。将此结果应用于三种常见镜映射，发现其对应的反向对称化规范均值分别为算术、log-Euclidean和调和均值。本工作深化了对现有对称化方法的理解，可作为选择合适均值的导航指南。
原文摘要 · Abstract (English)
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.
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