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.

Clustering above Exponential Families with Tempered Exponential Measures

“…Last, from a more general standpoint on TEMs, our general approach may seem close to the design of q-exponential families and even deformed exponential families -the knowledgeable reader will notice that our CODs technically look similar to escort distributions in the way we design them through (10), despite a normalization which belongs to the divisive normalisation of distribution rather than the subtractive normalisation of q-exponential families and deformed exponential families (Zhang and Wong, 2022) (alternatively, we rely on a t-subtractive normalisation, using the arithmetic of Nivanen et al (2003)). Classical escort distributions, however, appear independently of the qexponential families or deformed exponential families: they do not belong to their axiomatization.…”

Extended Divergence on a Foliation by Deformed Probability Simplexes

“…The application methods are open to consideration. It also remains to investigate the relationship with a new -duality on nonextensive statistical mechanics with mixed parameters [ 26 , 27 ].…”

When optimal transport meets information geometry

“…In the above theory, the κ λ function associated to c-duality, which originates as a cost function, is viewed as a deformation to the standard Legendre duality. This λ-deformation theory, fully developed in Zhang and Wong [ZW22], enables λ-deformed Legendre duality and the standard Legendre duality to be mutually transformable to each other based on a reparameterization of one of the λ-conjugate variables x, u. This fact underlies the finding of [WZ21], where a λ-deformed exponential family has two apparent expressions, i.e., one with subtractive normalization and the other with divisive normalization.…”

“…The former corresponds to the q-deformed exponential family associated to the Tsallis entropy and divergence, whereas the latter corresponds to the deformed exponential families studied by [81] associated to the Réyni entropy and divergence. The λ-deformation framework, inspired by the c-duality in optimal transport specializing the function form of κ λ , appears to provide a canonical extension of the dually flat geometry to an underlying manifold with constant curvature [81,84,90,91].…”