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A new group of transformations related to the Kullback–Leibler and Rényi divergences and universal classes of monotone measures of statistical complexity.

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Title: A new group of transformations related to the Kullback–Leibler and Rényi divergences and universal classes of monotone measures of statistical complexity.
Authors: Gabriel Iagar, Razvan1 (AUTHOR) razvan.iagar@urjc.es, Puertas-Centeno, David1,2 (AUTHOR) david.puertas@urjc.es, Toranzo, Elio V1 (AUTHOR) elio.vtoranzo@urjc.es
Source: Journal of Physics A: Mathematical & Theoretical. 2026, Vol. 59 Issue 26, p1-30. 30p.
Subjects: Probability density function, Transformation groups, Statistical measurement, Renyi's entropy, Information theory, Mathematical transformations
Abstract: In this work we introduce a family of transformations, named divergence transformations, interpolating between any pair of probability density functions sharing the same support. We prove the remarkable property that the whole family of Kullback–Leibler and Rényi divergences evolves in a monotone way with respect to the transformation parameter. Moreover, fixing the reference density, we show that the divergence transformations enjoy a group structure and can be derived through the algebraic conjugation of the differential-escort transformations and their relative counterparts. This algebraic structure allows us to deform any density function in such a way its divergence with respect to a fixed reference density might also increase as much as possible. We also establish the monotonicity of composed measures involving the proper Kullback–Leibler and Rényi divergences as well as other relative measures of moment and Fisher types. As applications, an approximation scheme of general density functions by simple functions is provided. In addition, we give a number of analytical and numerical examples of interest in both regimes of increasing and decreasing divergence. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:In this work we introduce a family of transformations, named divergence transformations, interpolating between any pair of probability density functions sharing the same support. We prove the remarkable property that the whole family of Kullback–Leibler and Rényi divergences evolves in a monotone way with respect to the transformation parameter. Moreover, fixing the reference density, we show that the divergence transformations enjoy a group structure and can be derived through the algebraic conjugation of the differential-escort transformations and their relative counterparts. This algebraic structure allows us to deform any density function in such a way its divergence with respect to a fixed reference density might also increase as much as possible. We also establish the monotonicity of composed measures involving the proper Kullback–Leibler and Rényi divergences as well as other relative measures of moment and Fisher types. As applications, an approximation scheme of general density functions by simple functions is provided. In addition, we give a number of analytical and numerical examples of interest in both regimes of increasing and decreasing divergence. [ABSTRACT FROM AUTHOR]
ISSN:17518113
DOI:10.1088/1751-8121/ae7c40