λ-Deformation: A Canonical Framework for Statistical Manifolds of Constant Curvature

Abstract

This paper systematically presents the λ-deformation as the canonical framework of deformation to the dually flat (Hessian) geometry, which has been well established in information geometry. We show that, based on deforming the Legendre duality, all objects in the Hessian case have their correspondence in the λ-deformed case: λ-convexity, λ-conjugation, λ-biorthogonality, λ-logarithmic divergence, λ-exponential and λ-mixture families, etc. In particular, λ-deformation unifies Tsallis and Rényi deformations by relating them to two manifestations of an identical λ-exponential family, under subtractive or divisive probability normalization, respectively. Unlike the different Hessian geometries of the exponential and mixture families, the λ-exponential family, in turn, coincides with the λ-mixture family after a change of random variables. The resulting statistical manifolds, while still carrying a dualistic structure, replace the Hessian metric and a pair of dually flat conjugate affine connections with a conformal Hessian metric and a pair of projectively flat connections carrying constant (nonzero) curvature. Thus, λ-deformation is a canonical framework in generalizing the well-known dually flat Hessian structure of information geometry.

Keywords: Legendre duality; conformal Hessian; constant curvature space; λ-duality; λ-exponential family; λ-mixture family.

Figure 1

Illustration of the $\lambda$-logarithmic divergence. Top: $\lambda =-1$ and $f\left(x\right)=\sqrt{x(10-x)}$. Bottom: $\lambda =1$ and $f\left(x\right)=2logx$. In both cases, $x^{\prime }=4$ and $x=8$, and we plot the function on the interval $(2,9)$. Note that the first-order logarithmic approximation (dashed grey curve) supports the graph of f from below.