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In many texts on information geometry one can find a proof that given a Bregman divergence, i.e. a divergence function of the form $$D_f(p\|q) = f(p)-f(q)+\nabla f(q)\cdot(p-q)$$ for some convex function $f$, there exists a canonical dually flat structure on the underlying (convex) set. Dually flat meaning that there exist two affine connections that are dual with respect to the metric induced by $D_f$.

Furthermore, these texts often also state and prove that the converse is true: given a dually flat manifold, there exists a canonical Bregman divergence inducing this structure. However, a general (dually) flat Riemannian manifold only admits affine charts in a local sense. These coordinate systems are not globally defined. Consequently I also expect that the convex potentials for the Bregman divergence only make sense locally.

Is this correct? In all texts they simply assume the existence of "an affine coordine system", but they never remark about the locality of this system. If this is correct I also wonder if on intersections of charts, the convex potentials have to coincide? (Since there exists some freedom.)

NDewolf
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Yes, a dually flat manifold only admits dual affine coordinate systems in a local sense. Furhtermore, the convex potential need not be unique, even for fixed dual affine coordinate systems.

However, the canonical divegence is independent on both the choice of affine coordinate systems and of convex potentials, which is the argument in Section 3.4 of Amari & Nagoka's Methods of Information Geometry. The degrees of freedom on the coordinate systems and on the potential cancel each other in the definition of the divergence.

Fábio M.
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