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Neural Computation

January 1, 2004, Vol. 16, No. 1, Pages 159-195
(doi: 10.1162/08997660460734047)
© 2003 Massachusetts Institute of Technology
Divergence Function, Duality, and Convex Analysis
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From a smooth, strictly convex function Φ: Rn → R, a parametric family of divergence function DΦ(α) may be introduced:

for x, y ∈ int dom(Φ) ⊂ Rn, and for α ∈ R, with DΦ(±1) defined through taking the limit of α. Each member is shown to induce an α-independent Riemannian metric, as well as a pair of dual α-connections, which are generally nonflat, except for α = ±1. In the latter case, D(±1)Φ reduces to the (nonparametric) Bregman divergence, which is representable using Φ and its convex conjugate Φ* and becomes the canonical divergence for dually flat spaces (Amari, 1982, 1985; Amari & Nagaoka, 2000). This formulation based on convex analysis naturally extends the information-geometric interpretation of divergence functions (Eguchi, 1983) to allow the distinction between two different kinds of duality: referential duality (α ⟷-α) and representational duality (Φ ⟷ Φ*). When applied to (not necessarily normalized) probability densities, the concept of conjugated representations of densities is introduced, so that ±α-connections defined on probability densities embody both referential and representational duality and are hence themselves bidual. When restricted to a finite-dimensional affine submanifold, the natural parameters of a certain representation of densities and the expectation parameters under its conjugate representation form biorthogonal coordinates. The alpha representation (indexed by β now, β [−1, 1]) is shown to be the only measure-invariant representation. The resulting two-parameter family of divergence functionals D(α, β), (α, β) [−1, 1] × [-1, 1] induces identical Fisher information but bidual alpha-connection pairs; it reduces in form to Amari's alpha-divergence family when α =±1 or when β = 1, but to the family of Jensen difference (Rao, 1987) when β = -1.