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

In this work we introduce a family of transformations, named \textit{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 recently introduced 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 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 recently introduced 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.

💡 Research Summary

The paper introduces a novel family of transformations, called divergence transformations, that interpolate between any two probability density functions (pdfs) sharing the same support. These transformations are constructed as algebraic conjugations of the previously defined differential‑escort transformations and their relative counterparts. As a result, the set of divergence transformations forms a group: the identity corresponds to the zero‑parameter transformation, the inverse is obtained by negating the parameter, and composition follows the usual additive rule for the parameter.

A central result is that both the Kullback‑Leibler (KL) divergence and the Rényi divergences evolve monotonically with respect to the transformation parameter. The authors prove this by first expressing the Rényi divergence in exponential form K_ξ