{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T22:12:17Z","timestamp":1777759937321,"version":"3.51.4"},"reference-count":20,"publisher":"MIT Press - Journals","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Neural Computation"],"published-print":{"date-parts":[[2004,1,1]]},"abstract":" From a smooth, strictly convex function \u03a6: Rn<\/jats:sup> \u2192 R, a parametric family of divergence function D\u03a6<\/jats:sub>(\u03b1)<\/jats:sup> may be introduced: <\/jats:p> [Formula: see text] <\/jats:p> for x, y, \u03b5 int dom(\u03a6) and for \u03b1 \u03b5 R, with D\u03a6<\/jats:sub>(\u00b11<\/jats:sup> defined through taking the limit of \u03b1. Each member is shown to induce an \u03b1-independent Riemannian metric, as well as a pair of dual \u03b1-connections, which are generally nonflat, except for \u03b1 = \u00b11. In the latter case, D(\u00b11)\u03a6<\/jats:sub> reduces to the (nonparametric) Bregman divergence, which is representable using and its convex conjugate \u03a6 * 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 (\u03b1 \uf578-\u03b1) and representational duality (\u03a6 \uf578 \u03a6 *). When applied to (not necessarily normalized) probability densities, the concept of conjugated representations of densities is introduced, so that \u00b1 \u03b1-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 \u03b2 now, \u03b2 \u03b5 [\u22121, 1]) is shown to be the only measure-invariant representation. The resulting two-parameter family of divergence functionals D(\u03b1, \u03b2)<\/jats:sup>, (\u03b1, \u03b2) \u03b5 [\u22121, 1] \u00d7 [-1, 1] induces identical Fisher information but bidual alpha-connection pairs; it reduces in form to Amari's alpha-divergence family when \u03b1 =\u00b11 or when \u03b2 = 1, but to the family of Jensen difference (Rao, 1987) when \u03b2 = 1. <\/jats:p>","DOI":"10.1162\/08997660460734047","type":"journal-article","created":{"date-parts":[[2004,1,21]],"date-time":"2004-01-21T22:31:06Z","timestamp":1074724266000},"page":"159-195","source":"Crossref","is-referenced-by-count":121,"title":["Divergence Function, Duality, and Convex Analysis"],"prefix":"10.1162","volume":"16","author":[{"given":"Jun","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Psychology, University of Michigan, Ann Arbor, MI 48109, U. S. 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