Symmetrizing Bregman divergences on positive definite matrices yields the arithmetic mean as canonical for forward symmetrization and the pulled-back dual arithmetic mean for reverse symmetrization, for any mirror map.
Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank,
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Symmetrizing Bregman Divergence on the Cone of Positive Definite Matrices: Which Mean to Use and Why
Symmetrizing Bregman divergences on positive definite matrices yields the arithmetic mean as canonical for forward symmetrization and the pulled-back dual arithmetic mean for reverse symmetrization, for any mirror map.