Divergence functions and geometric structures they induce on a manifold

Jun Zhang

pp. 1-30

in: Frank Nielsen (ed), Geometric theory of information, Berlin, Springer, 2014

Abstract

Divergence functions play a central role in information geometry. Given a manifold (mathfrak {M}), a divergence function(mathcal {D}) is a smooth, nonnegative function on the product manifold (mathfrak {M} imes mathfrak {M}) that achieves its global minimum of zero (with semi-positive definite Hessian) at those points that form its diagonal submanifold (varDelta _{mathfrak {M}} subset mathfrak {M} imes mathfrak {M}). In this chapter, we review how such divergence functions induce (i) a statistical structure (i.e., a Riemannian metric with a pair of conjugate affine connections) on (mathfrak {M}); (ii) a symplectic structure on (mathfrak {M} imes mathfrak {M}) if they are "proper"; (iii) a Kähler structure on (mathfrak {M} imes mathfrak {M}) if they further satisfy a certain condition. It is then shown that the class of (mathcal {D}_varPhi )-divergence functions [23], as induced by a strictly convex function(varPhi ) on (mathfrak {M}), satisfies all these requirements and hence makes (mathfrak {M} imes mathfrak {M}) a Kähler manifold (with Kähler potential given by (varPhi )). This provides a larger context for the (alpha )-Hessian structure induced by the (mathcal {D}_varPhi )-divergence on (mathfrak {M}), which is shown to be equiaffine admitting (alpha )-parallel volume forms and biorthogonal coordinates generated by (varPhi ) and its convex conjugate (varPhi ^{*}). As the (alpha )-Hessian structure is dually flat for (alpha = pm 1), the (mathcal {D}_varPhi )-divergence provides richer geometric structures (compared to Bregman divergence) to the manifold (mathfrak {M}) on which it is defined.