Divergence Function, Duality, and Convex Analysis

From a smooth, strictly convex function phi: Rn --> R, a parametric family of divergence function Dphi(alpha) may be introduced: equation: see text for x, y epsilon int dom (Phi) subset Rn, and for alpha in R, with Dphi(+/-1) defined through taking the limit of alpha. Each member is shown to induce an alpha-independent Riemannian metric, as well as a pair of dual alpha-connections, which are generally nonflat, except for alpha = +/-1. In the latter case, Dphi(+/-1) reduces to the (nonparametric) Bregman divergence, which is representable using phi and its convex conjugate phi* and becomes the canonical divergence for dually flat spaces (Amari, 1982, 1985; Amari it reduces in form to Amari's alpha-divergence family when alpha = +/-1 or when beta = 1, but to the family of Jensen difference (Rao, 1987) when beta = -1.