Problem
Research questions and friction points this paper is trying to address.
Bregman divergence
symmetrization
positive definite matrices
canonical mean
mirror map
Symmetrizing Bregman Divergence on the Cone of Positive Definite Matrices: Which Mean to Use and Why
🤖 AI Summary
This study addresses the symmetrization of Bregman divergences induced by mirror maps on the cone of positive definite matrices, with a focus on identifying appropriate symmetric means. By introducing a variational principle, the construction of symmetrized Bregman divergences is reformulated as a minimization problem over a class of mean functionals satisfying specific axioms. Theoretical analysis reveals that forward symmetrization corresponds to the arithmetic mean in the primal space, whereas reverse symmetrization corresponds to the pullback of the arithmetic mean from the dual space. This work provides the first systematic, axiomatically grounded and variational explanation for the choice of means, unifying several existing practices and explicitly deriving canonical mean expressions for three commonly used mirror maps.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Bregman divergence
symmetrization
positive definite matrices
mirror map
canonical mean
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