How type of Convexity of the Core function affects the Csisz\'{a}r $f$-divergence functional

TL;DR

This paper explores how different types of convexity of the core function influence the properties of the Csiszár $f$-divergence, including joint convexity and applications to inequalities and distances.

Contribution

It provides a comprehensive analysis of the impact of convexity types on the perspective functions and divergence measures, including new results on joint convexity and matrix Jensen inequalities.

Findings

Hellinger distance is jointly GG-convex.

Perspective functions' convexity depends on the core function and mean types.

Matrix Jensen inequality developed for various convexity types.

Abstract

We investigate how the type of Convexity of the Core function affects the Csisz\'{a}r $f$-divergence functional. A general treatment for the type of convexity has been considered and the associated perspective functions have been studied. In particular, it has been shown that when the core function is \rm{MN}-convex, then the associated perspective function is jointly \rm{MN}-convex if the two scalar mean \rm{M} and \rm{N} are the same. In the case where $\mathrm{M}\neq\mathrm{N}$, we study the type of convexity of the perspective function. As an application, we prove that the \textit{Hellinger distance} is jointly \rm{GG}-convex. As further applications, the matrix Jensen inequality has been developed for the perspective functions under different kinds of convexity.