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Divergence function, duality, and convex analysis.

Jun Zhang1

  • 1Department of Psychology, University of Michigan, Ann Arbor, MI 48109, USA. junz@umich.edu

Neural Computation
|March 10, 2004
PubMed
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This study introduces a generalized family of divergence functions derived from convex analysis, revealing dual connections and metrics applicable to probability densities. These bidual connections generalize existing divergence measures, offering new insights into information geometry.

Area of Science:

  • Information Geometry
  • Convex Analysis
  • Differential Geometry

Background:

  • Divergence functions are crucial in information geometry for measuring differences between probability distributions.
  • Existing frameworks like Bregman divergence and Amari's alpha-divergence have limitations in capturing dualities.
  • The need for a unified framework that encompasses various divergences and their dual properties is recognized.

Purpose of the Study:

  • To introduce a generalized parametric family of divergence functions, Dphi(alpha), from smooth, strictly convex functions.
  • To explore the induced Riemannian metrics and dual alpha-connections, analyzing their properties and special cases.
  • To extend the information-geometric interpretation of divergences by incorporating referential and representational dualities for probability densities.

Related Experiment Videos

Main Methods:

  • Derivation of a parametric family of divergence functions Dphi(alpha) from a convex function phi.
  • Analysis of the induced alpha-independent Riemannian metric and dual alpha-connections.
  • Application to probability densities, introducing conjugated representations and bidual connections.
  • Investigation of a two-parameter family of divergence functionals D(alpha,beta) on affine submanifolds.

Main Results:

  • Each Dphi(alpha) induces an alpha-independent Riemannian metric and a pair of dual alpha-connections.
  • For alpha = +/-1, Dphi(+/-1) reduces to Bregman divergence, canonical for dually flat spaces.
  • The framework naturally distinguishes between referential (alpha <--> -alpha) and representational (phi <--> phi*) dualities.
  • A two-parameter family D(alpha,beta) induces identical Fisher information but bidual alpha-connection pairs, generalizing known divergences.

Conclusions:

  • The proposed framework unifies and extends various divergence measures within information geometry.
  • The concept of bidual connections captures both referential and representational dualities inherent in convex analysis and probability theory.
  • This generalization provides a powerful tool for analyzing statistical manifolds and developing new statistical methods.