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About this paper
Ting-Kam Leonard Wong, Jun Zhang
A novel "λ-duality" is introduced to provide a unified mathematical framework for generalized exponential and mixture families, revealing that Rényi entropy and divergence naturally arise as central components within this extended information geometry. This framework establishes a consistent theoretical link between Tsallis and Rényi deformations, where λ represents the constant sectional curvature of the induced statistical manifold.
Key insights
- The λ-logarithmic divergence associated with the λ-potential function of the λ-exponential family is precisely the Rényi divergence of order q = 1-λ, and its λ-conjugate is the negative Rényi entropy.
- The framework provides a novel proof that the λ-exponential family maximizes Rényi entropy under escort expectation constraints, with Rényi divergence quantifying the optimality gap.
- The dual parameter derived from the λ-exponential family via the λ-gradient is identified as the unique right barycenter of the distribution with respect to the λ-logarithmic divergence, offering a new probabilistic interpretation.
Summary by alphaXiv.
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